50 Factoring Trinomials Worksheet Algebra 2 - Free Printable

Let’s solve each trinomial by factoring it into two binomials: (x + m)(x + n), where m and n are numbers that multiply to the constant term (c) and add up to the middle coefficient (b).

We’ll go one by one.

---

1. x² + 13x + 12

Find two numbers that multiply to 12 and add to 13 → 1 and 12
→ Factors: (x + 1)(x + 12)

Left factor: (x + 1) → matches pattern with triangle symbol → look at key: (x+1) is paired with letter A? Wait — let’s check the key carefully.

Looking at the key:

The left column has patterns like (x+1), (x+2)... and each has a shaded shape next to it. The right column has (x-1), (x-2)... and letters A, B, C...

But the instruction says:
> find one factor in the left column (how to color it) and one factor in the right column (which letter to color).

Wait — actually, looking again:

The grid on the left has letters. Then there’s a key with shapes for left-column factors (like (x+1), (x+2), etc.) and letters for right-column factors (like (x-1)=A, (x-2)=B, etc.)

But our trinomials may have both positive and negative factors.

Actually, re-reading:

“Factor the trinomials and find one factor in the left column (how to color it) and one factor in the right column (which letter to color).”

So for each factored form, say (x + a)(x + b), we need to match:

- One of the factors to the LEFT column list (which uses shapes to indicate how to color)
- The other factor to the RIGHT column list (which assigns a letter to color)

But note: the left column lists only forms like (x+1), (x+2), ..., (x+12), (x-9) — wait, actually looking:

Left column:
(x + 1) → triangle
(x + 2) → square
...
(x + 12) → ?
Also includes (x - 9) → triangle? Wait no — let me list them properly from image description:

From your text:

Left column (with shapes):
(x + 1) → △
(x + 2) ■
(x + 3) ▽
(x + 4) ◊
(x + 5) △ (again? maybe different shading)
Actually, since this is text-based, perhaps the shapes are just indicators — but for solving, we don’t need the shapes yet. We need to factor correctly first.

Right column (letters):
(x - 1) → A
(x - 2) → B
(x - 3) → C
(x - 4) → D
(x - 5) → E
(x - 6) → F
(x + 7) → G ← note: this is PLUS!
(x + 8) → H
(x + 9) → I
(x + 10) → J
(x + 11) → K
(x + 12) → L

Wait — so the right column includes some positive terms too! Specifically:

G = (x + 7)
H = (x + 8)
I = (x + 9)
J = (x + 10)
K = (x + 11)
L = (x + 12)

And negatives:
A = (x - 1)
B = (x - 2)
C = (x - 3)
D = (x - 4)
E = (x - 5)
F = (x - 6)

What about (x + 1) to (x + 6)? They’re in the left column with shapes.

Similarly, (x - 7) to (x - 12) are in left column? Let's see:

Left column also has:
(x - 7) → ▽
(x - 8) → △
(x - 9) → ▽
(x - 10) → □
(x - 11) → ▨
(x - 12) → ▧
(x - 9) again? Probably typo in transcription.

Actually, from your original text:

Left column entries:
(x + 1)
(x + 2)
(x + 3)
(x + 4)
(x + 5)
(x + 6)
(x - 7)
(x - 8)
(x - 9)
(x - 10)
(x - 11)
(x - 12)
(x - 9) ← duplicate? Maybe error.

But importantly, for factoring, we can proceed algebraically.

Let’s factor all 12 trinomials first.

---

Problem 1: x² + 13x + 12

Factors of 12 that add to 13: 1 and 12 → (x + 1)(x + 12)

Now, which goes to left column? Left column has (x+1), (x+2), ..., (x+6), and also (x-7) to (x-12). So (x+1) is in left column.

(x+12) is in right column → assigned letter L.

So for this one: left factor = (x+1) → use its shape; right factor = (x+12) → color letter L.

But we don’t need to output colors — just solve the factoring. Since the question is to “solve the problem”, and the problem is to factor and presumably identify which letters/shapes to color, but since user didn’t ask for coloring, maybe just factor?

Wait — the user said: “Solve the problem accurately.” And the problem is to factor the trinomials and find the matching factors in the columns.

But since the final answer should be concise, and options are false, probably just provide the factored forms.

But let’s confirm with another.

---

Problem 2: x² + 14x + 40

Factors of 40 that add to 14: 4 and 10 → (x + 4)(x + 10)

(x + 4) is in left column (has a shape)
(x + 10) is in right column → letter J

---

Problem 3: x² + 13x + 42

Factors of 42 that add to 13: 6 and 7 → (x + 6)(x + 7)

(x + 6) → left column
(x + 7) → right column → letter G

---

Problem 4: x² - 11x + 28

Need two numbers that multiply to 28, add to -11 → both negative: -4 and -7 → (x - 4)(x - 7)

(x - 4) → right column → letter D
(x - 7) → left column (has shape)

Note: order doesn't matter for multiplication, but for matching, we assign one to left, one to right.

In this case, (x - 7) is in left column, (x - 4) is in right column → D

---

Problem 5: x² - 13x + 40

Multiply to 40, add to -13 → -5 and -8 → (x - 5)(x - 8)

(x - 5) → right column → E
(x - 8) → left column

---

Problem 6: x² - 16x + 60

Multiply to 60, add to -16 → -6 and -10 → (x - 6)(x - 10)

(x - 6) → right column → F
(x - 10) → left column

---

Problem 7: x² + x - 2

Multiply to -2, add to 1 → 2 and -1 → (x + 2)(x - 1)

(x + 2) → left column
(x - 1) → right column → A

---

Problem 8: x² + x - 6

Multiply to -6, add to 1 → 3 and -2 → (x + 3)(x - 2)

(x + 3) → left
(x - 2) → right → B

---

Problem 9: x² + 2x - 15

Multiply to -15, add to 2 → 5 and -3 → (x + 5)(x - 3)

(x + 5) → left
(x - 3) → right → C

---

Problem 10: x² - 3x - 88

Multiply to -88, add to -3 → factors: 8 and -11? 8 * -11 = -88, 8 + (-11) = -3 → yes → (x + 8)(x - 11)

(x + 8) → right column → H
(x - 11) → left column

Wait — (x + 8) is in right column as H, (x - 11) is in left column.

Yes.

---

Problem 11: x² - 3x - 108

Multiply to -108, add to -3 → need two numbers: let's think... 9 and -12? 9*(-12)= -108, 9 + (-12)= -3 → yes → (x + 9)(x - 12)

(x + 9) → right column → I
(x - 12) → left column

---

Problem 12: x² + 2x - 99

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

(x + 11) → right column → K
(x - 9) → left column

Perfect.

Now, to present the final answer — since the task is to factor and match, but the user likely wants the factored forms, and perhaps the corresponding letters? But the instruction says "solve the problem", and in context, probably list the factored forms.

But to be precise, let's see what the expected output is.

Since the worksheet has a grid to color based on factors, but we're not doing the coloring, I think providing the factored form for each is sufficient.

Alternatively, maybe list for each trinomial the two factors and which is left/right, but that might be overkill.

Given the constraints, I'll provide the factored forms in order.

Final Answer:

1. (x + 1)(x + 12)
2. (x + 4)(x + 10)
3. (x + 6)(x + 7)
4. (x - 4)(x - 7)
5. (x - 5)(x - 8)
6. (x - 6)(x - 10)
7. (x + 2)(x - 1)
8. (x + 3)(x - 2)
9. (x + 5)(x - 3)
10. (x + 8)(x - 11)
11. (x + 9)(x - 12)
12. (x + 11)(x - 9)

Double-checking calculations:

1. 1*12=12, 1+12=13 ✓
2. 4*10=40, 4+10=14 ✓
3. 6*7=42, 6+7=13 ✓
4. (-4)*(-7)=28, -4-7=-11 ✓
5. (-5)*(-8)=40, -5-8=-13 ✓
6. (-6)*(-10)=60, -6-10=-16 ✓
7. 2*(-1)=-2, 2-1=1 ✓
8. 3*(-2)=-6, 3-2=1 ✓
9. 5*(-3)=-15, 5-3=2 ✓
10. 8*(-11)=-88, 8-11=-3 ✓
11. 9*(-12)=-108, 9-12=-3 ✓
12. 11*(-9)=-99, 11-9=2 ✓

All correct.

Final Answer:
1. (x + 1)(x + 12)
2. (x + 4)(x + 10)
3. (x + 6)(x + 7)
4. (x - 4)(x - 7)
5. (x - 5)(x - 8)
6. (x - 6)(x - 10)
7. (x + 2)(x - 1)
8. (x + 3)(x - 2)
9. (x + 5)(x - 3)
10. (x + 8)(x - 11)
11. (x + 9)(x - 12)
12. (x + 11)(x - 9)