High School Math Worksheets | Math Worksheets PDF - Free Printable

Since the user has not selected any of the options (ExplainAnswer, TeachTopic, GeneratePractice), I will provide the solutions to all the problems listed in the image as a comprehensive answer key. This is the most helpful response given the context of "helping a student solve homework problems."

I will go through each section and solve the quadratic expressions by factoring.

Section A: Simple Quadratics (Coefficient of x² is 1)
To factor these, we look for two numbers that multiply to the last number and add to the middle number.

1) x² + 7x - 30
* Multiply to -30, add to 7.
* Numbers: +10, -3
* Answer: (x + 10)(x - 3)

2) x² + 9x + 20
* Multiply to 20, add to 9.
* Numbers: +4, +5
* Answer: (x + 4)(x + 5)

3) x² + 8x - 9
* Multiply to -9, add to 8.
* Numbers: +9, -1
* Answer: (x + 9)(x - 1)

4) x² - 18x + 80
* Multiply to 80, add to -18.
* Numbers: -8, -10
* Answer: (x - 8)(x - 10)

5) x² - 11x + 28
* Multiply to 28, add to -11.
* Numbers: -4, -7
* Answer: (x - 4)(x - 7)

6) x² + 6x - 72
* Multiply to -72, add to 6.
* Numbers: +12, -6
* Answer: (x + 12)(x - 6)

7) x² - 9x - 22
* Multiply to -22, add to -9.
* Numbers: -11, +2
* Answer: (x - 11)(x + 2)

8) x² - x - 12
* Multiply to -12, add to -1.
* Numbers: -4, +3
* Answer: (x - 4)(x + 3)

9) x² + 3x - 108
* Multiply to -108, add to 3.
* Numbers: +12, -9 (Wait, 12*-9 = -108, 12-9=3. Correct.)
* Answer: (x + 12)(x - 9)

10) x² - 17x + 72
* Multiply to 72, add to -17.
* Numbers: -8, -9
* Answer: (x - 8)(x - 9)

11) x² - x - 42
* Multiply to -42, add to -1.
* Numbers: -7, +6
* Answer: (x - 7)(x + 6)

12) x² - 15x + 56
* Multiply to 56, add to -15.
* Numbers: -7, -8
* Answer: (x - 7)(x - 8)

Section B: Coefficient of x² is 2
We need to find factors where the outer and inner terms add up to the middle term. Alternatively, use the "ac method" (multiply a*c, find factors, split middle term).

1) 2x² + 3x + 1
* (2x + 1)(x + 1) -> 2x + x = 3x. Correct.
* Answer: (2x + 1)(x + 1)

2) 2x² + 5x + 2
* (2x + 1)(x + 2) -> 4x + x = 5x. Correct.
* Answer: (2x + 1)(x + 2)

3) 2x² + 7x + 3
* (2x + 1)(x + 3) -> 6x + x = 7x. Correct.
* Answer: (2x + 1)(x + 3)

4) 2x² + 7x + 5
* (2x + 5)(x + 1) -> 2x + 5x = 7x. Correct.
* Answer: (2x + 5)(x + 1)

5) 2x² + 9x + 7
* (2x + 7)(x + 1) -> 2x + 7x = 9x. Correct.
* Answer: (2x + 7)(x + 1)

6) 2x² + 5x + 3
* (2x + 3)(x + 1) -> 2x + 3x = 5x. Correct.
* Answer: (2x + 3)(x + 1)

7) 2x² + 8x + 6
* First, factor out 2: 2(x² + 4x + 3)
* Then factor inside: 2(x + 3)(x + 1)
* Answer: 2(x + 3)(x + 1)

8) 2x² + 9x + 10
* (2x + 5)(x + 2) -> 4x + 5x = 9x. Correct.
* Answer: (2x + 5)(x + 2)

9) 2x² + 16x + 14
* Factor out 2: 2(x² + 8x + 7)
* Factor inside: 2(x + 7)(x + 1)
* Answer: 2(x + 7)(x + 1)

10) 2x² + 16x + 24
* Factor out 2: 2(x² + 8x + 12)
* Factor inside: 2(x + 6)(x + 2)
* Answer: 2(x + 6)(x + 2)

11) 2x² + 12x + 18
* Factor out 2: 2(x² + 6x + 9)
* Factor inside: 2(x + 3)(x + 3) or 2(x+3)²
* Answer: 2(x + 3)(x + 3)

12) 2x² + 14x + 20
* Factor out 2: 2(x² + 7x + 10)
* Factor inside: 2(x + 5)(x + 2)
* Answer: 2(x + 5)(x + 2)

13) 2x² + 22x + 36
* Factor out 2: 2(x² + 11x + 18)
* Factor inside: 2(x + 9)(x + 2)
* Answer: 2(x + 9)(x + 2)

14) 2x² + 28x + 48
* Factor out 2: 2(x² + 14x + 24)
* Factor inside: 2(x + 12)(x + 2)
* Answer: 2(x + 12)(x + 2)

15) 2x² + 26x + 72
* Factor out 2: 2(x² + 13x + 36)
* Factor inside: 2(x + 9)(x + 4)
* Answer: 2(x + 9)(x + 4)

Section C: Mixed Coefficients (2, 3, 5)

1) 2x² + x - 1
* (2x - 1)(x + 1) -> 2x - x = x. Correct.
* Answer: (2x - 1)(x + 1)

2) 2x² + x - 3
* (2x + 3)(x - 1) -> -2x + 3x = x. Correct.
* Answer: (2x + 3)(x - 1)

3) 2x² + 9x - 5
* (2x - 1)(x + 5) -> 10x - x = 9x. Correct.
* Answer: (2x - 1)(x + 5)

4) 2x² - 3x - 2
* (2x + 1)(x - 2) -> -4x + x = -3x. Correct.
* Answer: (2x + 1)(x - 2)

5) 2x² - 13x - 24
* (2x + 3)(x - 8) -> -16x + 3x = -13x. Correct.
* Answer: (2x + 3)(x - 8)

6) 3x² - 14x - 5
* (3x + 1)(x - 5) -> -15x + x = -14x. Correct.
* Answer: (3x + 1)(x - 5)

7) 3x² - 8x - 11
* (3x - 11)(x + 1) -> 3x - 11x = -8x. Correct.
* Answer: (3x - 11)(x + 1)

8) 2x² - 14x + 12
* Factor out 2: 2(x² - 7x + 6)
* Factor inside: 2(x - 6)(x - 1)
* Answer: 2(x - 6)(x - 1)

9) 3x² - 21x + 36
* Factor out 3: 3(x² - 7x + 12)
* Factor inside: 3(x - 4)(x - 3)
* Answer: 3(x - 4)(x - 3)

10) 5x² - 41x + 8
* (5x - 1)(x - 8) -> -40x - x = -41x. Correct.
* Answer: (5x - 1)(x - 8)

11) 3x² - 2x - 21
* (3x - 7)(x + 3) -> 9x - 7x = 2x (Sign error check: needs -2x).
* Try (3x + 7)(x - 3) -> -9x + 7x = -2x. Correct.
* Answer: (3x + 7)(x - 3)

12) 2x² + 2x - 12
* Factor out 2: 2(x² + x - 6)
* Factor inside: 2(x + 3)(x - 2)
* Answer: 2(x + 3)(x - 2)

13) 2x² - 11x + 15
* (2x - 5)(x - 3) -> -6x - 5x = -11x. Correct.
* Answer: (2x - 5)(x - 3)

14) 3x² - 34x - 24
* (3x + 2)(x - 12) -> -36x + 2x = -34x. Correct.
* Answer: (3x + 2)(x - 12)

15) 5x² - 27x + 10
* (5x - 2)(x - 5) -> -25x - 2x = -27x. Correct.
* Answer: (5x - 2)(x - 5)

Section D: Higher Coefficients and Challenges

1) 4x² + 12x + 8
* Factor out 4: 4(x² + 3x + 2)
* Factor inside: 4(x + 2)(x + 1)
* Answer: 4(x + 2)(x + 1)

2) 6x² + 23x + 20
* (2x + 5)(3x + 4) -> 8x + 15x = 23x. Correct.
* Answer: (2x + 5)(3x + 4)

3) 6x² + 24x + 18
* Factor out 6: 6(x² + 4x + 3)
* Factor inside: 6(x + 3)(x + 1)
* Answer: 6(x + 3)(x + 1)

4) 4x² + 31x + 21
* (4x + 3)(x + 7) -> 28x + 3x = 31x. Correct.
* Answer: (4x + 3)(x + 7)

5) 4x² + 22x + 18
* Factor out 2: 2(2x² + 11x + 9)
* Factor inside: 2(2x + 9)(x + 1)
* Answer: 2(2x + 9)(x + 1)

6) 8x² + 22x + 15
* (2x + 3)(4x + 5) -> 10x + 12x = 22x. Correct.
* Answer: (2x + 3)(4x + 5)

7) 8x² + 36x + 16
* Factor out 4: 4(2x² + 9x + 4)
* Factor inside: 4(2x + 1)(x + 4)
* Answer: 4(2x + 1)(x + 4)

8) 6x² - 5x - 4
* (2x + 1)(3x - 4) -> -8x + 3x = -5x. Correct.
* Answer: (2x + 1)(3x - 4)

9) 6x² - 16x + 15
* Discriminant: (-16)² - 4(6)(15) = 256 - 360 = -104. No real roots. Cannot be factored over integers.
* Wait, let me re-check the image. Ah, problem 9 is `6x² - 16x + 15`.
* Let's check factors of 90 (6*15) that subtract to 16.
* Factors of 90: 1,90; 2,45; 3,30; 5,18; 6,15; 9,10. None differ by 16.
* Is it possible the image says something else? Looking closely at crop 5... it looks like `6x² - 16x + 15`.
* Let me check if I misread a number. Maybe it's `6x² - 16x + 8`? No, looks like 15.
* Maybe `6x² - 19x + 15`? (3x-5)(2x-3) -> -9x-10x=-19x.
* Maybe `6x² - 13x + 6`? (2x-3)(3x-2).
* Let's assume the question is correct as written: `6x² - 16x + 15`. It is prime over integers.
* *Self-correction*: Let me look at the full image again. Problem 9 in Section D is `6x² - 16x + 15`.
* Actually, let me check problem 16 in Section D: `18x² + 51x + 8`.
* Let me re-examine problem 9. Is it possible it's `6x² - 16x + 8`? If so, 2(3x² - 8x + 4) = 2(3x-2)(x-2).
* Is it possible it's `6x² - 11x + 4`? (2x-1)(3x-4).
* Let's look at the spacing. The '1' in 16 is close to the '6'. The '1' in 15 is close to the '5'.
* Let's try to solve `6x² - 16x + 15` using quadratic formula just to be sure. x = [16 ± sqrt(256 - 360)] / 12. Complex roots.
* Perhaps it is `6x² - 16x + 10`? 2(3x² - 8x + 5) = 2(3x-5)(x-1).
* Perhaps it is `6x² - 19x + 15`? (3x-5)(2x-3).
* Let's assume there might be a typo in the worksheet or I am misreading. Let's look really closely at the original image.
* Problem 9: `6x² - 16x + 15`. It definitely looks like 15.
* Wait, let's look at problem 10: `8x² + 16x - 24`. Factor out 8: 8(x² + 2x - 3) = 8(x+3)(x-1).
* Let's look at problem 9 again. Maybe it's `6x² - 16x + 6`? 2(3x² - 8x + 3). No.
* Maybe it's `6x² - 16x + 8`? 2(3x² - 8x + 4) = 2(3x-2)(x-2).
* Given the context of these worksheets, it's highly likely to be factorable. The most likely intended problem is `6x² - 16x + 8` or `6x² - 19x + 15`.
* However, I must solve what is written. If it's not factorable, I will state that.
* *Alternative*: Maybe it is `6x² - 16x + 10`? 2(3x-5)(x-1).
* Let's look at the font. The '1' in 16 and 15 look identical.
* Let's try one more combination. `6x² - 16x + 15`.
* Okay, I will note that this one appears to be prime (not factorable with integers) as written, but I'll double check my arithmetic. 6*15=90. Factors of 90: 1,2,3,5,6,9,10,15,18,30,45,90. Sums/Diffs: 1+90=91, 90-1=89... 9+10=19. 6+15=21. 5+18=23. 3+30=33. 2+45=47. 1+90=91.
* None of the factor pairs of 90 add or subtract to 16. So, strictly speaking, it cannot be factored over integers.
* *Hypothesis*: Could it be `6x² - 16x + 10`? (Factors of 60: 6,10 -> sum 16). Yes! 6x² - 6x - 10x + 10 = 6x(x-1) -10(x-1) = (6x-10)(x-1) = 2(3x-5)(x-1).
* Could it be `6x² - 16x + 8`? (Factors of 48: 4,12 -> sum 16). Yes! 6x² - 4x - 12x + 8 = 2x(3x-2) -4(3x-2) = (2x-4)(3x-2) = 2(x-2)(3x-2).
* Given the visual similarity, `6x² - 16x + 8` seems plausible if the 5 is actually an 8? No, it looks very much like a 5.
* Let's assume the question is `6x² - 16x + 15` and it's a "trick" question or a typo. I will provide the answer as "Cannot be factored over integers" but add a note.
* *Wait*, looking at problem 16: `18x² + 51x + 8`. 18*8=144. Factors of 144 summing to 51? 3, 48. 3+48=51. Yes! (18x + 3)(x + 8) -> 3(6x+1)(x+8). Or (6x+?)(3x+?).
* Let's go back to 9. I'll just solve it as written and note it's prime.
* *Correction*: I will look at the provided solution in my head. If I assume it's `6x² - 16x + 10`, it works perfectly. If I assume `6x² - 16x + 8`, it works perfectly.
* Let's look at the spacing again. `16x + 15`. The gap between + and 15 is small.
* Okay, I will list it as Prime (Not factorable) but mention the likely typo.
* *Actually*, let's look at problem 19: `36x² + 42x - 8`. Factor out 2: 2(18x² + 21x - 4). 18*-4 = -72. Factors of -72 summing to 21? 24, -3. Yes. 2(18x² + 24x - 3x - 4) = 2(6x(3x+4) -1(3x+4)) = 2(6x-1)(3x+4).
* Okay, I'm confident in my ability to factor the others. For #9, I'll state it's not factorable.

10) 8x² + 16x - 24
* Factor out 8: 8(x² + 2x - 3)
* Factor inside: 8(x + 3)(x - 1)
* Answer: 8(x + 3)(x - 1)

11) 7x² + 52x - 32
* (7x - 4)(x + 8) -> 56x - 4x = 52x. Correct.
* Answer: (7x - 4)(x + 8)

12) 6x² - 55x + 56
* (2x - 7)(3x - 8) -> -16x - 21x = -37x (No).
* (6x - 7)(x - 8) -> -48x - 7x = -55x. Correct.
* Answer: (6x - 7)(x - 8)

13) 9x² + 36x + 35
* (3x + 5)(3x + 7) -> 21x + 15x = 36x. Correct.
* Answer: (3x + 5)(3x + 7)

14) 12x² - 42x + 30
* Factor out 6: 6(2x² - 7x + 5)
* Factor inside: 6(2x - 5)(x - 1)
* Answer: 6(2x - 5)(x - 1)

15) 48x² + 72x + 24
* Factor out 24: 24(2x² + 3x + 1)
* Factor inside: 24(2x + 1)(x + 1)
* Answer: 24(2x + 1)(x + 1)

16) 18x² + 51x + 8
* (3x + 8)(6x + 1) -> 3x + 48x = 51x. Correct.
* Answer: (3x + 8)(6x + 1)

17) 20x² + 27x + 9
* (4x + 3)(5x + 3) -> 12x + 15x = 27x. Correct.
* Answer: (4x + 3)(5x + 3)

18) 30x² - 32x + 8
* Factor out 2: 2(15x² - 16x + 4)
* Factor inside: 2(3x - 2)(5x - 2) -> -6x - 10x = -16x. Correct.
* Answer: 2(3x - 2)(5x - 2)

19) 36x² + 42x - 8
* Factor out 2: 2(18x² + 21x - 4)
* Factor inside: 2(6x - 1)(3x + 4) -> 24x - 3x = 21x. Correct.
* Answer: 2(6x - 1)(3x + 4)

20) 24x² - 50x - 14
* Factor out 2: 2(12x² - 25x - 7)
* Factor inside: 2(3x - 7)(4x + 1) -> 3x - 28x = -25x. Correct.
* Answer: 2(3x - 7)(4x + 1)

21) -2x² + x + 21 (Challenge!)
* Factor out -1 first to make it easier: -(2x² - x - 21)
* Factor inside: -(2x + 7)(x - 3) -> -6x + 7x = x. Correct.
* Answer: -(2x + 7)(x - 3) or (2x + 7)(3 - x)

Final check of Section D #9. I will assume it's a typo for `6x² - 16x + 8` or `6x² - 16x + 10` but since I have to provide an answer, I will state it is not factorable over integers as written. However, looking at the pattern of the worksheet, it's almost certainly a typo. I'll provide the answer for `6x² - 16x + 8` as a likely intended solution but clarify.
Actually, let's look at `6x² - 16x + 15` again. Is there any way? No.
Let's assume the user just wants the answers. I will provide the answers for all clearly solvable ones and note the issue with #9.

Wait, looking at the image again very carefully.
Problem 9: `6x² - 16x + 15`.
Is it possible the middle term is `19x`? `6x² - 19x + 15` -> (3x-5)(2x-3).
Is it possible the last term is `10`? `6x² - 16x + 10` -> 2(3x-5)(x-1).
Is it possible the last term is `8`? `6x² - 16x + 8` -> 2(3x-2)(x-2).
Given the visual evidence, I will treat it as unsolvable but mention the likely typo.

Let's do a final review of the list.
Section A: All good.
Section B: All good.
Section C: All good.
Section D:
1-8: Good.
9: Problematic.
10-21: Good.

I will formulate the response now. I will list the answers by section.

Final Answer Construction:
I will present the answers clearly, section by section.
For Section D #9, I will write "Prime (Not factorable with integers)" and add a small note that there might be a typo in the question.

One detail: Section D #21 is marked "Challenge!". I've solved it as `-(2x+7)(x-3)`.

Let's double check Section D #12: `6x² - 55x + 56`.
(6x - 7)(x - 8) = 6x² - 48x - 7x + 56 = 6x² - 55x + 56. Correct.

Section D #16: `18x² + 51x + 8`.
(3x + 8)(6x + 1) = 18x² + 3x + 48x + 8 = 18x² + 51x + 8. Correct.

Section D #19: `36x² + 42x - 8`.
2(18x² + 21x - 4).
18*-4 = -72. Factors of -72 adding to 21: 24, -3.
18x² + 24x - 3x - 4
6x(3x + 4) - 1(3x + 4)
(6x - 1)(3x + 4).
So 2(6x - 1)(3x + 4). Correct.

Okay, ready to write.

Structure:
I will provide the answers in a list format corresponding to the sections in the image.

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

Section B
1) (2x + 1)(x + 1)
2) (2x + 1)(x + 2)
3) (2x + 1)(x + 3)
4) (2x + 5)(x + 1)
5) (2x + 7)(x + 1)
6) (2x + 3)(x + 1)
7) 2(x + 3)(x + 1)
8) (2x + 5)(x + 2)
9) 2(x + 7)(x + 1)
10) 2(x + 6)(x + 2)
11) 2(x + 3)(x + 3)
12) 2(x + 5)(x + 2)
13) 2(x + 9)(x + 2)
14) 2(x + 12)(x + 2)
15) 2(x + 9)(x + 4)

Section C
1) (2x - 1)(x + 1)
2) (2x + 3)(x - 1)
3) (2x - 1)(x + 5)
4) (2x + 1)(x - 2)
5) (2x + 3)(x - 8)
6) (3x + 1)(x - 5)
7) (3x - 11)(x + 1)
8) 2(x - 6)(x - 1)
9) 3(x - 4)(x - 3)
10) (5x - 1)(x - 8)
11) (3x + 7)(x - 3)
12) 2(x + 3)(x - 2)
13) (2x - 5)(x - 3)
14) (3x + 2)(x - 12)
15) (5x - 2)(x - 5)

Section D
1) 4(x + 2)(x + 1)
2) (2x + 5)(3x + 4)
3) 6(x + 3)(x + 1)
4) (4x + 3)(x + 7)
5) 2(2x + 9)(x + 1)
6) (2x + 3)(4x + 5)
7) 4(2x + 1)(x + 4)
8) (2x + 1)(3x - 4)
9) Note: As written ($6x^2 - 16x + 15$), this cannot be factored with integers. It is likely a typo for $6x^2 - 16x + 8$ which would be $2(3x-2)(x-2)$.
10) 8(x + 3)(x - 1)
11) (7x - 4)(x + 8)
12) (6x - 7)(x - 8)
13) (3x + 5)(3x + 7)
14) 6(2x - 5)(x - 1)
15) 24(2x + 1)(x + 1)
16) (3x + 8)(6x + 1)
17) (4x + 3)(5x + 3)
18) 2(3x - 2)(5x - 2)
19) 2(6x - 1)(3x + 4)
20) 2(3x - 7)(4x + 1)
21) -(2x + 7)(x - 3)

I will format this cleanly. I will omit the detailed explanation of the typo unless necessary, but since I'm acting as an assistant, pointing out the potential error is helpful. I'll just list the answer as "Not factorable (likely typo)" to be concise.

Wait, looking at Section D #9 again. Is it possible it is `6x² - 16x + 10`?
If I look at the spacing, the '1' in 16 and 15 are aligned. The '5' is distinct.
I'll stick with "Not factorable".

One final check on Section C #11: `3x² - 2x - 21`.
(3x - 7)(x + 3) -> 9x - 7x = 2x. Wrong sign.
(3x + 7)(x - 3) -> -9x + 7x = -2x. Correct.
Answer: (3x + 7)(x - 3). Correct.

One final check on Section D #21: `-2x² + x + 21`.
Factor out -1: `-(2x² - x - 21)`.
Factors of -21 adding to -1: -7, +3.
`-(2x - 7)(x + 3)` -> `-(2x² + 6x - 7x - 21)` -> `-(2x² - x - 21)` -> `-2x² + x + 21`.
Wait.
My previous calculation: `-(2x + 7)(x - 3)`.
Let's expand `-(2x + 7)(x - 3)`.
`(2x + 7)(x - 3) = 2x² - 6x + 7x - 21 = 2x² + x - 21`.
Negate it: `-2x² - x + 21`.
The original problem is `-2x² + x + 21`.
So my previous calculation was wrong.
Let's re-solve Section D #21.
`-2x² + x + 21`
Factor out -1: `-(2x² - x - 21)`
We need factors of -21 that add to -1 (the coefficient of x in `2x² - x - 21`).
Factors of -21: (-7, 3), (7, -3), (-21, 1), (21, -1).
-7 + 3 = -4.
7 + (-3) = 4.
-21 + 1 = -20.
21 + (-1) = 20.
None of these add to -1.
Wait.
Let's use the quadratic formula for `2x² - x - 21 = 0`.
x = [1 ± sqrt(1 - 4(2)(-21))] / 4
x = [1 ± sqrt(1 + 168)] / 4
x = [1 ± sqrt(169)] / 4
x = [1 ± 13] / 4
x1 = 14/4 = 7/2
x2 = -12/4 = -3
So the factors are (x - 7/2) and (x + 3).
To get integer coefficients: (2x - 7) and (x + 3).
So `2x² - x - 21 = (2x - 7)(x + 3)`.
Therefore, `-2x² + x + 21 = -(2x - 7)(x + 3)`.
Let's check: `-(2x² + 6x - 7x - 21) = -(2x² - x - 21) = -2x² + x + 21`. Correct.
So the answer is `-(2x - 7)(x + 3)` or `(7 - 2x)(x + 3)`.

Okay, correction made.

Final plan:
List all answers clearly.
Add a note for Section D #9.
Add the corrected answer for Section D #21.

One last check on Section D #12: `6x² - 55x + 56`.
(6x - 7)(x - 8) = 6x² - 48x - 7x + 56 = 6x² - 55x + 56. Correct.

Okay, everything seems solid. I will generate the response now.
The user wants me to solve the problem. Since it's a whole sheet, I will provide the answer key.

Response Structure:
I will organize the answers by Section (A, B, C, D) to match the worksheet.
I will use bold for the final answers.
I will add a small note about the unsolvable problem in Section D.

Let's double check Section B #7: `2x² + 8x + 6`.
Factor out 2: `2(x² + 4x + 3)`.
Factors of 3 adding to 4: 1, 3.
`2(x + 1)(x + 3)`. Correct.

Section B #11: `2x² + 12x + 18`.
Factor out 2: `2(x² + 6x + 9)`.
Factors of 9 adding to 6: 3, 3.
`2(x + 3)(x + 3)`. Correct.

Section C #8: `2x² - 14x + 12`.
Factor out 2: `2(x² - 7x + 6)`.
Factors of 6 adding to -7: -1, -6.
`2(x - 1)(x - 6)`. Correct.

Section C #9: `3x² - 21x + 36`.
Factor out 3: `3(x² - 7x + 12)`.
Factors of 12 adding to -7: -3, -4.
`3(x - 3)(x - 4)`. Correct.

Section D #1: `4x² + 12x + 8`.
Factor out 4: `4(x² + 3x + 2)`.
Factors of 2 adding to 3: 1, 2.
`4(x + 1)(x + 2)`. Correct.

Section D #5: `4x² + 22x + 18`.
Factor out 2: `2(2x² + 11x + 9)`.
Factors of 18 (2*9) adding to 11: 2, 9.
`2(2x² + 2x + 9x + 9) = 2(2x(x+1) + 9(x+1)) = 2(2x+9)(x+1)`. Correct.

Section D #7: `8x² + 36x + 16`.
Factor out 4: `4(2x² + 9x + 4)`.
Factors of 8 (2*4) adding to 9: 1, 8.
`4(2x² + x + 8x + 4) = 4(x(2x+1) + 4(2x+1)) = 4(x+4)(2x+1)`. Correct.

All checks passed. Proceeding with response generation.
Here are the solutions for all the quadratic expressions in the worksheet, organized by section.

Section A

*These are simple quadratics where the $x^2$ term has a coefficient of 1.*

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

Section B

*These quadratics have a coefficient of 2 for the $x^2$ term. Remember to check if you can factor out a common number first.*

1) $(2x + 1)(x + 1)$
2) $(2x + 1)(x + 2)$
3) $(2x + 1)(x + 3)$
4) $(2x + 5)(x + 1)$
5) $(2x + 7)(x + 1)$
6) $(2x + 3)(x + 1)$
7) $2(x + 3)(x + 1)$
8) $(2x + 5)(x + 2)$
9) $2(x + 7)(x + 1)$
10) $2(x + 6)(x + 2)$
11) $2(x + 3)(x + 3)$
12) $2(x + 5)(x + 2)$
13) $2(x + 9)(x + 2)$
14) $2(x + 12)(x + 2)$
15) $2(x + 9)(x + 4)$

Section C

*These have mixed coefficients (2, 3, 5) for the $x^2$ term.*

1) $(2x - 1)(x + 1)$
2) $(2x + 3)(x - 1)$
3) $(2x - 1)(x + 5)$
4) $(2x + 1)(x - 2)$
5) $(2x + 3)(x - 8)$
6) $(3x + 1)(x - 5)$
7) $(3x - 11)(x + 1)$
8) $2(x - 6)(x - 1)$
9) $3(x - 4)(x - 3)$
10) $(5x - 1)(x - 8)$
11) $(3x + 7)(x - 3)$
12) $2(x + 3)(x - 2)$
13) $(2x - 5)(x - 3)$
14) $(3x + 2)(x - 12)$
15) $(5x - 2)(x - 5)$

Section D

*These are more complex problems with higher coefficients. Some require factoring out a common number first.*

1) $4(x + 2)(x + 1)$
2) $(2x + 5)(3x + 4)$
3) $6(x + 3)(x + 1)$
4) $(4x + 3)(x + 7)$
5) $2(2x + 9)(x + 1)$
6) $(2x + 3)(4x + 5)$
7) $4(2x + 1)(x + 4)$
8) $(2x + 1)(3x - 4)$
9) Note: As written ($6x^2 - 16x + 15$), this cannot be factored using integers. It is likely a typo in the worksheet. If it were $6x^2 - 16x + 8$, the answer would be $2(3x - 2)(x - 2)$.
10) $8(x + 3)(x - 1)$
11) $(7x - 4)(x + 8)$
12) $(6x - 7)(x - 8)$
13) $(3x + 5)(3x + 7)$
14) $6(2x - 5)(x - 1)$
15) $24(2x + 1)(x + 1)$
16) $(3x + 8)(6x + 1)$
17) $(4x + 3)(5x + 3)$
18) $2(3x - 2)(5x - 2)$
19) $2(6x - 1)(3x + 4)$
20) $2(3x - 7)(4x + 1)$
21) $-(2x - 7)(x + 3)$ *(Challenge problem)*