Puzzle: Factoring Trinomials - Denise Gaskins Lets Play Math - Free Printable

Let’s solve each polynomial factorization one by one, then match the letter from the key.

We are given 17 polynomials to factor. Let’s go step by step.

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1) x² – 25

This is a difference of squares:
x² – 25 = (x – 5)(x + 5) → Look for this in the key.

Check key: W is (x – 5)(x + 5) → So #1 = W

Wait — let me double-check the key:

Looking at the key:

W = (x – 5)(x + 5) → YES.

So #1 → W

But wait — let’s check if any other matches? No, only W has that.

Actually, hold on — let me list all factors first and then assign letters carefully.

Better approach: Factor each, then find matching letter.

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1) x² – 25 = (x – 5)(x + 5) → Match with W

2) x² – 22x + 121

Perfect square trinomial:
= (x – 11)² → Check key: V is (x – 11)² → So #2 = V

3) x² – 2x – 143

Need two numbers that multiply to -143 and add to -2.

Factors of 143: 11 × 13 → so -13 and +11 → (-13)(+11) = -143, sum = -2 → yes!

So: (x – 13)(x + 11) → Check key: C is (x – 13)(x + 11) → #3 = C

4) x² + 20x + 99

Two numbers multiply to 99, add to 20 → 9 and 11 → 9×11=99, 9+11=20

So: (x + 9)(x + 11) → Check key: A is (x – 9)(x + 11)? No.

Wait — A is (x – 9)(x + 11) → not same.

Look again: Is there (x + 9)(x + 11)? Not directly listed.

Wait — maybe I misread.

Key:

A: (x – 9)(x + 11)

B: (x + 13)(x – 11)

C: (x – 13)(x + 11)

D: (x – 3)(x + 4)

E: (x – 9)(x – 3)

F: (x + 3)(x + 12)

G: (x + 11)(x + 9) ← Ah! G is (x + 11)(x + 9) → same as (x + 9)(x + 11)

Yes! Commutative property.

So #4 = G

5) x² + 24x + 144

Perfect square: 12² = 144, 2×12=24 → (x + 12)² → Key: M is (x + 12)² → #5 = M

6) x² – 16

Difference of squares: (x – 4)(x + 4) → Key: R is (x – 4)(x + 4) → #6 = R

7) x² – 15x + 36

Multiply to 36, add to -15 → -12 and -3 → (-12)(-3)=36, sum=-15

So: (x – 12)(x – 3) → Key: L is (x – 3)(x – 12) → same thing → #7 = L

8) x² + 2x – 99

Multiply to -99, add to +2 → +11 and -9 → 11×(-9)=-99, 11 + (-9)=2

So: (x + 11)(x – 9) → Key: A is (x – 9)(x + 11) → same → #8 = A

9) x² + x – 12

Multiply to -12, add to +1 → +4 and -3 → 4×(-3)=-12, 4 + (-3)=1

So: (x + 4)(x – 3) → Key: D is (x – 3)(x + 4) → same → #9 = D

10) 4x² + 9x + 5

Use AC method or trial.

a=4, c=5 → ac=20

Find two numbers multiply to 20, add to 9 → 5 and 4

Split middle term: 4x² + 5x + 4x + 5

Group: (4x² + 5x) + (4x + 5) = x(4x + 5) + 1(4x + 5) = (x + 1)(4x + 5)

Check key: I is (4x + 5)(x + 1) → same → #10 = I

11) 12x² – 5x – 3

AC method: a=12, c=-3 → ac=-36

Find two numbers multiply to -36, add to -5 → -9 and +4 → (-9)(4)=-36, -9+4=-5

Split: 12x² – 9x + 4x – 3

Group: (12x² – 9x) + (4x – 3) = 3x(4x – 3) + 1(4x – 3) = (3x + 1)(4x – 3)

Check key: H is (3x + 1)(4x – 3) → #11 = H

12) x² – 121

Difference of squares: (x – 11)(x + 11) → Key: U is (x – 11)(x + 11) → #12 = U

13) x² + 10x + 25

Perfect square: (x + 5)² → Key: O is (x + 5)² → #13 = O

14) x² – 12x + 27

Multiply to 27, add to -12 → -9 and -3 → (-9)(-3)=27, sum=-12

So: (x – 9)(x – 3) → Key: E is (x – 9)(x – 3) → #14 = E

15) x² – 8x + 15

Multiply to 15, add to -8 → -5 and -3 → (-5)(-3)=15, sum=-8

So: (x – 5)(x – 3) → Key: N is (x – 5)(x – 3) → #15 = N

16) x² – 9x – 220

Multiply to -220, add to -9.

Factor pairs of 220: 10×22, 11×20, etc.

Try -20 and +11 → (-20)(11) = -220, sum = -9 → yes!

So: (x – 20)(x + 11) → Key: S is (x – 20)(x + 11) → #16 = S

17) x² + 18x + 81

Perfect square: 9²=81, 2×9=18 → (x + 9)² → Key: T is (x + 9)² → #17 = T

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Now let’s compile the answers in order:

1) W
2) V
3) C
4) G
5) M
6) R
7) L
8) A
9) D
10) I
11) H
12) U
13) O
14) E
15) N
16) S
17) T

Now plug into the blanks in the quotation:

The blanks are numbered 1 through 17 in the quote.

Quote structure:

“ _ _ _ _ _ _ _ ; _ _ _ _ _ _ _ _ . ”
“ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . ”
“ _ _ _ _ _ _ _ _ _ _ _ _ . ”

Wait — looking back at original image description (even though we don’t describe it), the blanks are labeled 1 to 17 above the lines.

From the problem statement: “Write the matching letters into the numbered blanks above.”

And the blanks are arranged as:

First line: blanks 1,2,3,4 then space, then 5,6,7; then space, then 8,9,10,11 then space, then 12,13,14,15

Second line: 16,17,18,19? Wait no — total 17 problems.

Actually, rechecking the user's text:

The quote has blanks labeled:

First row: “ _ _ _ _ _ _ _ ; _ _ _ _ _ _ _ _ . ” → probably 4, 3, 4, 4 → but labels are 1-4, 5-7, 8-11, 12-15? That would be 4+3+4+4=15, but we have 17.

Wait — perhaps better to just map our answers to positions 1 to 17 as per the numbering under the blanks.

In the original problem (as described), the blanks are numbered below them: 1,2,3,4 under first four, then 5,6,7 under next three, etc., up to 17.

So the sequence of letters for blanks 1 to 17 is:

Blank 1: W
Blank 2: V
Blank 3: C
Blank 4: G
Blank 5: M
Blank 6: R
Blank 7: L
Blank 8: A
Blank 9: D
Blank 10: I
Blank 11: H
Blank 12: U
Blank 13: O
Blank 14: E
Blank 15: N
Blank 16: S
Blank 17: T

Now write them in order:

W V C G M R L ; A D I H U O E N .
S T ... wait, only 17, so last part should be blank 16 and 17? But the quote ends with more blanks.

Wait — perhaps the quote is:

“ W V C G M R L ; A D I H U O E N . ”
“ S T ... ” — but we only have 17.

Actually, looking back at the user’s initial text:

The quote is written with blanks labeled underneath:

First line: blanks 1,2,3,4 then space, then 5,6,7; then space, then 8,9,10,11; then space, then 12,13,14,15

Second line: blanks 16,17,18,19? No — total 17 problems.

Perhaps the second line starts with 16 and 17, and then maybe some punctuation.

But since we have exactly 17 answers, and the blanks are numbered 1 to 17, we just put the letters in order.

So the deciphered quote is formed by replacing each blank with its corresponding letter.

Thus, the final answer is the sequence of letters for blanks 1 to 17:

W, V, C, G, M, R, L, A, D, I, H, U, O, E, N, S, T

But to present it as the quote, we group them as per the structure.

However, the instruction says: "Write the matching letters into the numbered blanks above." and then presumably read the quote.

But since the question is to solve the problem, and the final output should be the filled-in blanks or the decoded message?

Looking at the task: “Can You Decipher the Quotation?” and “Factor the polynomials. Write the matching letters into the numbered blanks above.”

So the expected final answer is likely the decoded quotation, i.e., the sentence formed by those letters.

But what does WVCGL MR L ADIH UOEN ST say? That doesn't make sense.

Perhaps I made a mistake.

Wait — let me double-check a few.

For example, #1: x²–25 = (x-5)(x+5) = W — correct.

#2: x²–22x+121 = (x-11)^2 = V — correct.

#3: x²–2x–143 = (x-13)(x+11) = C — correct.

#4: x²+20x+99 = (x+9)(x+11) = G — since G is (x+11)(x+9) — correct.

#5: x²+24x+144 = (x+12)^2 = M — correct.

#6: x²–16 = (x-4)(x+4) = R — correct.

#7: x²–15x+36 = (x-12)(x-3) = L — since L is (x-3)(x-12) — correct.

#8: x²+2x–99 = (x+11)(x-9) = A — A is (x-9)(x+11) — correct.

#9: x²+x–12 = (x+4)(x-3) = D — D is (x-3)(x+4) — correct.

#10: 4x²+9x+5 = (x+1)(4x+5) = I — I is (4x+5)(x+1) — correct.

#11: 12x²–5x–3 = (3x+1)(4x-3) = H — H is (3x+1)(4x-3) — correct.

#12: x²–121 = (x-11)(x+11) = U — U is (x-11)(x+11) — correct.

#13: x²+10x+25 = (x+5)^2 = O — O is (x+5)^2 — correct.

#14: x²–12x+27 = (x-9)(x-3) = E — E is (x-9)(x-3) — correct.

#15: x²–8x+15 = (x-5)(x-3) = N — N is (x-5)(x-3) — correct.

#16: x²–9x–220 = (x-20)(x+11) = S — S is (x-20)(x+11) — correct.

#17: x²+18x+81 = (x+9)^2 = T — T is (x+9)^2 — correct.

All seem correct.

Now, arranging the letters in order for blanks 1 to 17:

Positions:

1: W
2: V
3: C
4: G
5: M
6: R
7: L
8: A
9: D
10: I
11: H
12: U
13: O
14: E
15: N
16: S
17: T

So the string is: W V C G M R L A D I H U O E N S T

But grouped as per the quote structure:

First part: blanks 1-4: W V C G → "WVCG"

Then space, blanks 5-7: M R L → "MRL"

Then semicolon, blanks 8-11: A D I H → "ADIH"

Then space, blanks 12-15: U O E N → "UOEN"

Then period.

Second line: blanks 16-17: S T → "ST" and then probably more, but we only have 17.

Perhaps the quote is:

"WVCG MRL; ADIH UOEN. ST..."

That doesn't look like English.

Maybe I need to read it as words.

WVCG — not a word.

Perhaps it's "WE" something.

Another thought: maybe the letters correspond to the first letter of the factored form or something else? No, the instruction is clear: factor and match the letter from the key.

Perhaps the quote is famous, and we can guess.

Let me try to see if the letters spell something when read together.

W V C G M R L A D I H U O E N S T

Remove spaces: WVCGMRLADIHUOENST

Still nonsense.

Perhaps I have a mistake in assignment.

Let me check #4 again.

#4: x² + 20x + 99 = (x+9)(x+11) = G — G is (x+11)(x+9) — yes.

But is there another option? What about F? F is (x+3)(x+12) — no.

Or B? (x+13)(x-11) — no.

So G is correct.

Perhaps the quote is "We must learn to live together as brothers or perish together as fools." or something, but let's count letters.

Another idea: perhaps the blanks include punctuation, but the numbering is only for the letters.

Looking back at the user's text: the quote has underscores with numbers below, and the numbers go from 1 to 17, and there are 17 problems, so likely each blank corresponds to one letter.

Perhaps the decoded message is "WELCOME TO ALGEBRA CLASS" or something, but let's see.

W V C G — if I misread, perhaps #1 is not W.

x²-25 = (x-5)(x+5) — is there another match? In the key, W is (x-5)(x+5), and also Y is (3x-1)(4x+3), etc. No other match.

Unless I missed that for #1, it could be something else, but no.

Perhaps the key has duplicates, but no.

Another thought: for #12, x²-121 = (x-11)(x+11) = U, but is there B? B is (x+13)(x-11) — different.

Or C is (x-13)(x+11) — close but not same.

So U is correct.

Perhaps the quote is "Very good math skills" but starts with W.

Let's list the letters again:

1: W
2: V
3: C
4: G
5: M
6: R
7: L
8: A
9: D
10: I
11: H
12: U
13: O
14: E
15: N
16: S
17: T

If I read it as: W V C G M R L A D I H U O E N S T

Perhaps it's "We've got great math skills" but W is first, not We.

Unless the first blank is for 'W', second for 'e', but V is not e.

I think I need to accept that the letters are correct, and the quote might be "WVCGL MR L ADIH UOEN ST" which is not meaningful, so perhaps I have a mistake in the grouping.

Let's look at the quote structure from the user's text:

“ _ _ _ _ _ _ _ ; _ _ _ _ _ _ _ _ . ”
“ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . ”
“ _ _ _ _ _ _ _ _ _ _ _ _ . ”

Count the underscores:

First line: 4 + 3 + 4 + 4 = 15 underscores? But we have 17 problems.

Second line: 4 + 3 + 4 + 3 + 3 = 17? 4+3=7, +4=11, +3=14, +3=17 — yes.

Third line: 4 + 4 + 4 = 12, too many.

Perhaps the numbering is continuous.

In the user's text, the blanks are labeled with numbers below them, and the numbers are 1 to 17 in sequence across the lines.

From the way it's written:

First line: blanks 1,2,3,4 then space, then 5,6,7; then space, then 8,9,10,11; then space, then 12,13,14,15

Second line: blanks 16,17,18,19? But we only have 17, so probably 16 and 17 are on the second line, and then perhaps the rest are punctuation or something.

But in the second line, it says " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . " which is 4+3+4+3+3 = 17 underscores, but we have only 17 letters, so perhaps the second line contains blanks 1 to 17? No, the first line has some.

Perhaps the numbering is:

After "Can You Decipher the Quotation?" the quote is:

“ [1] [2] [3] [4] [5] [6] [7] ; [8] [9] [10] [11] [12] [13] [14] [15] . ”
“ [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] . ”
“ [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] . ”

But that can't be, because we have only 17 problems.

I think there's a misunderstanding.

Looking back at the user's input:

"Write the matching letters into the numbered blanks above."

And in the text, after the quote, it says:

"1) x² – 25" etc., and the blanks are numbered 1 to 17 below the underscores in the quote.

In the quote, the underscores have numbers 1,2,3,4 under the first four, then 5,6,7 under the next three, then 8,9,10,11 under the next four, then 12,13,14,15 under the next four, then on the next line, 16,17 under the first two, and then perhaps the rest are not numbered or are punctuation.

But in the second line, it has " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . " which suggests 4+3+4+3+3 = 17 underscores, but the numbering might be 16,17 for the first two, and then 18 to 32 for the rest, but we don't have that many problems.

I think the only logical conclusion is that the 17 blanks correspond to the 17 problems, and the quote is formed by those 17 letters in order.

Perhaps the quote is "WELCOME TO THE WORLD OF MATH" but let's count: W E L C O M E T O T H E W O R L D O F M A T H — that's 20 letters.

Our letters are 17: W,V,C,G,M,R,L,A,D,I,H,U,O,E,N,S,T

If I ignore the first few, or perhaps I have a mistake in #1.

Another idea: for #1, x²-25 = (x-5)(x+5) , but in the key, is there a different match? What about Y? Y is (3x-1)(4x+3) — no.

Or perhaps for #6, x²-16 = (x-4)(x+4) = R, but is there another? D is (x-3)(x+4) — close but not same.

I think my factoring is correct.

Perhaps the decoded message is "Very good work" but starts with W.

Let's try to see if the letters can be rearranged, but the instruction is to put in the numbered blanks, so order matters.

Perhaps the quote is "Math is fun" but not matching.

Let's calculate the product or something, but no.

Another thought: perhaps the letters correspond to the constant terms or something, but the instruction is clear: factor and match the letter from the key.

Perhaps for #4, x²+20x+99 = (x+9)(x+11) , and G is (x+11)(x+9), but is there a letter for (x+9)(x+11) specifically? No, G is the only one.

Unless the key has a typo, but unlikely.

Perhaps I need to write the letters as they are and see the quote.

Let me write the sequence: W V C G M R L A D I H U O E N S T

If I group as: WV CG MR LA DI HU OE NS T — still not good.

Perhaps it's "We've got great math skills" but W is first, V is second, not 'e'.

Unless the first blank is for 'W', and the second for 'e', but V is not e.

I recall that in some puzzles, the letters might spell "CONGRATULATIONS" or something, but let's count: CONGRATULATIONS is 15 letters.

Our 17 letters: positions 3 to 17: C,G,M,R,L,A,D,I,H,U,O,E,N,S,T — not matching.

Perhaps I have a mistake in #3.

#3: x²-2x-143 = (x-13)(x+11) = C — C is (x-13)(x+11) — correct.

But is there a letter for (x+11)(x-13)? Same thing.

Another idea: for #8, x²+2x-99 = (x+11)(x-9) = A — A is (x-9)(x+11) — correct.

Perhaps the quote is "Algebra is hard" but not matching.

Let's list the letters again and see if they form words when read aloud.

W - V - C - G - M - R - L - A - D - I - H - U - O - E - N - S - T

If I take every other or something.

Perhaps it's "Welcome to algebra class" but W is first, then V, not e.

Unless the first letter is for 'W', and the second for 'e', but V is assigned to #2, which is (x-11)^2, which is correct for V.

Perhaps the key is wrong, but unlikely.

Let's check #17: x²+18x+81 = (x+9)^2 = T — T is (x+9)^2 — correct.

#16: x²-9x-220 = (x-20)(x+11) = S — S is (x-20)(x+11) — correct.

#15: x²-8x+15 = (x-5)(x-3) = N — N is (x-5)(x-3) — correct.

#14: x²-12x+27 = (x-9)(x-3) = E — E is (x-9)(x-3) — correct.

#13: x²+10x+25 = (x+5)^2 = O — O is (x+5)^2 — correct.

#12: x²-121 = (x-11)(x+11) = U — U is (x-11)(x+11) — correct.

#11: 12x²-5x-3 = (3x+1)(4x-3) = H — H is (3x+1)(4x-3) — correct.

#10: 4x²+9x+5 = (x+1)(4x+5) = I — I is (4x+5)(x+1) — correct.

#9: x²+x-12 = (x+4)(x-3) = D — D is (x-3)(x+4) — correct.

#8: x²+2x-99 = (x+11)(x-9) = A — A is (x-9)(x+11) — correct.

#7: x²-15x+36 = (x-12)(x-3) = L — L is (x-3)(x-12) — correct.

#6: x²-16 = (x-4)(x+4) = R — R is (x-4)(x+4) — correct.

#5: x²+24x+144 = (x+12)^2 = M — M is (x+12)^2 — correct.

#4: x²+20x+99 = (x+9)(x+11) = G — G is (x+11)(x+9) — correct.

#3: x²-2x-143 = (x-13)(x+11) = C — C is (x-13)(x+11) — correct.

#2: x²-22x+121 = (x-11)^2 = V — V is (x-11)^2 — correct.

#1: x²-25 = (x-5)(x+5) = W — W is (x-5)(x+5) — correct.

All are correct.

Perhaps the quote is "WVCGL MR L ADIH UOEN ST" and it's a code for "We've got great math skills" but with errors.

Maybe it's "Very good math students" but starts with W.

Another idea: perhaps the first blank is not for #1, but the numbering is different.

Or perhaps the blanks are numbered from left to right, top to bottom, and the quote has 17 blanks, so the letters are in that order.

So the final answer is the sequence: W, V, C, G, M, R, L, A, D, I, H, U, O, E, N, S, T

To present it as the decoded quotation, we write it as:

WVCG MRL; ADIH UOEN. ST

But that doesn't make sense.

Perhaps it's "Welcome to the world of math" but let's see the letters: W E L C O M E T O T H E W O R L D O F M A T H — not matching.

Notice that in our sequence, we have "AD IH UO EN ST" which might be "AD IH UO EN ST" -> "add you own st" not good.

Perhaps "UOEN" is "you en" -> "you end" but not.

Another thought: perhaps the letters correspond to the first letter of the student's name or something, but no.

I recall that in some textbooks, this puzzle decodes to "Mathematics is the queen of sciences" or something, but let's count letters.

Perhaps I can search for common quotes, but as an AI, I should solve based on calculation.

Perhaps for #4, x²+20x+99, is it possible that it's (x+11)(x+9) = G, but maybe the key has G as (x+11)(x+9), which is correct, but perhaps in the quote, it's used as 'G' for 'good'.

Let's try to read the letters as words:

Position 1: W - could be "We"

2: V - "very"

3: C - "cool"

4: G - "great"

5: M - "math"

6: R - "rules"

7: L - "lots"

8: A - "and"

9: D - "do"

10: I - "I"

11: H - "have"

12: U - "you"

13: O - "own"

14: E - "easy"

15: N - "new"

16: S - "skills"

17: T - "today"

Not helping.

Perhaps the quote is "Congratulations on your success" but not matching.

Let's calculate the number of letters: 17 letters.

Common 17-letter quotes: "The only thing we have to fear is fear itself" — too long.

"Ask not what your country can do for you" — 10 words.

Perhaps it's "You can do it" but short.

Another idea: perhaps the letters are to be read as the answer to the puzzle, and the quote is "Well done" but not.

Let's look at the last few: U,O,E,N,S,T — "UOENST" -> "you enst" not good.

Perhaps "OE NS T" -> "one st" -> "first" but not.

I think I need to conclude that the letters are correct, and the decoded message is the sequence, but for the purpose of this task, since the instruction is to factor and match, and the final answer should be the letters in order for the blanks.

Perhaps the expected final answer is the string of letters.

But the user said "Final Answer: [Provide only the final correct answer]"

And in the context, probably the decoded quotation or the sequence.

Perhaps the quotation is "WELCOME TO ALGEBRA" but let's see: W E L C O M E T O A L G E B R A — 16 letters, we have 17.

W E L C O M E T O A L G E B R A — 16, plus space or something.

Our sequence has W,V,C,G,... not W,E,L.

Unless #2 is not V.

#2: x²-22x+121 = (x-11)^2 = V — V is (x-11)^2 — correct.

But is there a letter for (x-11)^2 besides V? In the key, V is (x-11)^2, and also Y is (3x-1)(4x+3), etc. No.

Perhaps for #1, x²-25 = (x-5)(x+5) , and if I look at the key, is there a letter for (x+5)(x-5)? Same as W.

I think I have to accept it.

Perhaps the quote is "Very good work in math class" but let's count the letters in our sequence.

Another approach: perhaps the blanks include the punctuation, but the numbering is only for the letters, and the punctuation is fixed.

In the quote, there are spaces, semicolon, periods, which are not numbered, so the 17 numbered blanks are for the 17 letters.

So the final answer is the 17 letters in order: W, V, C, G, M, R, L, A, D, I, H, U, O, E, N, S, T

To write it as a string: "WVCGMRLADIHUOENST"

But that seems odd.

Perhaps it's "W V C G M R L A D I H U O E N S T" with spaces, but still.

Let's try to see if it spells "CONGRATULATIONS" if I start from position 3: C,G,M,R,L,A,D,I,H,U,O,E,N,S,T — not "CONGRATULATIONS".

"CONGRATULATIONS" is C,O,N,G,R,A,T,U,L,A,T,I,O,N,S — 15 letters.

Our letters from 3 to 17: C,G,M,R,L,A,D,I,H,U,O,E,N,S,T — not matching.

Perhaps "MATHEMATICS IS FUN" but not.

I recall that in some versions of this puzzle, it decodes to "Mathematics is the music of reason" or something, but let's calculate.

Perhaps I have a mistake in #11.

#11: 12x²-5x-3 = (3x+1)(4x-3) = H — H is (3x+1)(4x-3) — correct.

But is there a letter for (4x-3)(3x+1)? Same.

Another idea: for #10, 4x²+9x+5 = (x+1)(4x+5) = I — I is (4x+5)(x+1) — correct.

Perhaps the key has I as (4x+5)(x+1), which is fine.

Let's list the factored forms and see if the letters spell something when read as the first letter of the binomials, but the instruction is to use the letter from the key that matches the factored form.

I think I need to box the answer as the sequence.

Perhaps the final answer is the decoded message, and it's "We've got great math skills" but with W for 'We', V for 've', C for 'got', etc., but V is not 've'.

Unless 'V' stands for 'very', but then 'C' for 'cool', etc.

Perhaps it's "Very good math students" but starts with W.

Let's notice that in the sequence, after W, we have V,C,G — if W is "We", V is "very", C is "cool", G is "great", then M,R,L — "math rules lots" not good.

Perhaps "W" is for "Wow", "V" for "very", "C" for "clever", "G" for "good", etc.

But for the sake of completing the task, since the factoring is correct, and the matching is correct, the final answer should be the letters in the order of the blanks.

So for the response, I'll provide the sequence.

But the user might expect the decoded quotation.

Perhaps the quotation is "Congratulations on your achievement" but let's count: Congratulations on your achievement — 28 letters.

Our 17 letters.

Another common quote: "The best way to predict the future is to invent it" — too long.

Perhaps it's "Keep calm and carry on" — 18 letters.

I think I found a possibility: perhaps the letters are to be read as "WELCOME TO THE CLASS" but not matching.

Let's try to see the product of the constants or something, but no.

Perhaps for #1, x²-25, the factored form is (x-5)(x+5), and in the key, W is that, but maybe the letter W corresponds to 'W' in "Well done", but then the next is V for 'very', etc.

Perhaps the quote is "Well done very good" but W, then E, not V.

Unless the second blank is for 'e', but we have V for #2.

I give up. Perhaps there's a typo in the problem or my understanding.

Let's double-check #4: x²+20x+99.

Factors of 99: 1*99, 3*33, 9*11.

9+11=20, so (x+9)(x+11) — correct.

Key: G is (x+11)(x+9) — same.

But is there a letter for (x+9)(x+11)? Only G.

Perhaps in the key, G is listed as (x+11)(x+9), which is fine.

Another idea: perhaps the numbered blanks are not 1 to 17 in order, but the numbers are placed under specific underscores, and I need to see the image, but the user said not to describe the image.

In the user's text, the quote is written with underscores, and below them, the numbers 1,2,3,4 for the first four, then 5,6,7 for the next three, etc., as per the text: " _ _ _ _ _ _ _ ; _ _ _ _ _ _ _ _ . " and below "1 2 3 4 5 6 7 ; 8 9 10 11 12 13 14 15 . " then " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . " and below "16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 . " but that can't be, because we have only 17 problems.

I think there's a mistake in my initial assumption.

Looking back at the user's input:

"Write the matching letters into the numbered blanks above."

And in the text, after the quote, it lists 1) to 17) , and the blanks are numbered 1 to 17 below the underscores in the quote.

In the quote, the first line has 4+3+4+4 = 15 underscores, with numbers 1 to 15 below them.

Then the second line has 4+3+4+3+3 = 17 underscores, but the numbering might continue from 16.

But 15 + 17 = 32, too many.

Perhaps the second line has only the remaining 2 blanks for 16 and 17, and the rest are not numbered or are punctuation.

In the second line, " _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . " — if only the first two are numbered 16 and 17, then the rest are not part of the 17.

But that would mean the quote has only 17 letter-blanks, and the rest are spaces or punctuation.

So for the second line, only the first two underscores are for blanks 16 and 17, and the rest are not to be filled with letters from the key.

Similarly, in