Factoring trinomials practice worksheet for algebra students, focusing on expressions where the leading coefficient is greater than 1.

Here's the complete solution for factoring each trinomial where the leading coefficient $ a > 1 $. We'll use the AC method (also called splitting the middle term) for most, and factor out GCFs when possible.

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1. $ 4x^2 + 4x - 35 $

- Multiply $ a \cdot c = 4 \cdot (-35) = -140 $
- Find two numbers that multiply to -140 and add to 4 → 14 and -10
- Split middle term:
$ 4x^2 + 14x - 10x - 35 $
- Group:
$ (4x^2 + 14x) + (-10x - 35) $
- Factor each group:
$ 2x(2x + 7) -5(2x + 7) $
- Factor out common binomial:
$ (2x + 7)(2x - 5) $

✔ Answer: $ \boxed{(2x + 7)(2x - 5)} $

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2. $ 5x^2 + 37x - 24 $

- $ a \cdot c = 5 \cdot (-24) = -120 $
- Two numbers: multiply to -120, add to 37 → 40 and -3
- Split:
$ 5x^2 + 40x - 3x - 24 $
- Group:
$ (5x^2 + 40x) + (-3x - 24) $
- Factor:
$ 5x(x + 8) -3(x + 8) $
- Binomial:
$ (5x - 3)(x + 8) $

✔ Answer: $ \boxed{(5x - 3)(x + 8)} $

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3. $ 3x^2 + 12x + 12 $

- First, factor out GCF: 3
$ 3(x^2 + 4x + 4) $
- Now factor inside:
$ x^2 + 4x + 4 = (x + 2)^2 $
- So overall:
$ 3(x + 2)^2 $

✔ Answer: $ \boxed{3(x + 2)^2} $

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4. $ 6x^2 - 44x + 14 $

- GCF of all terms is 2:
$ 2(3x^2 - 22x + 7) $
- Now factor $ 3x^2 - 22x + 7 $
- $ a \cdot c = 3 \cdot 7 = 21 $; need two numbers that multiply to 21 and add to -22 → -21 and -1
- Split:
$ 3x^2 - 21x - x + 7 $
- Group:
$ (3x^2 - 21x) + (-x + 7) $
- Factor:
$ 3x(x - 7) -1(x - 7) $
- Binomial:
$ (3x - 1)(x - 7) $
- Include the 2:
$ 2(3x - 1)(x - 7) $

✔ Answer: $ \boxed{2(3x - 1)(x - 7)} $

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5. $ 3x^2 - 23x - 36 $

- $ a \cdot c = 3 \cdot (-36) = -108 $
- Numbers: multiply to -108, add to -23 → -27 and 4
- Split:
$ 3x^2 - 27x + 4x - 36 $
- Group:
$ (3x^2 - 27x) + (4x - 36) $
- Factor:
$ 3x(x - 9) + 4(x - 9) $
- Binomial:
$ (3x + 4)(x - 9) $

✔ Answer: $ \boxed{(3x + 4)(x - 9)} $

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6. $ 2x^2 - 3x + 1 $

- $ a \cdot c = 2 \cdot 1 = 2 $
- Numbers: multiply to 2, add to -3 → -2 and -1
- Split:
$ 2x^2 - 2x - x + 1 $
- Group:
$ (2x^2 - 2x) + (-x + 1) $
- Factor:
$ 2x(x - 1) -1(x - 1) $
- Binomial:
$ (2x - 1)(x - 1) $

✔ Answer: $ \boxed{(2x - 1)(x - 1)} $

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7. $ 20x^2 + 2x - 6 $

- GCF is 2:
$ 2(10x^2 + x - 3) $
- Factor $ 10x^2 + x - 3 $
- $ a \cdot c = 10 \cdot (-3) = -30 $
- Numbers: multiply to -30, add to 1 → 6 and -5
- Split:
$ 10x^2 + 6x - 5x - 3 $
- Group:
$ (10x^2 + 6x) + (-5x - 3) $
- Factor:
$ 2x(5x + 3) -1(5x + 3) $
- Binomial:
$ (2x - 1)(5x + 3) $
- Include 2:
$ 2(2x - 1)(5x + 3) $

✔ Answer: $ \boxed{2(2x - 1)(5x + 3)} $

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8. $ 3x^2 + 25x + 42 $

- $ a \cdot c = 3 \cdot 42 = 126 $
- Numbers: multiply to 126, add to 25 → 18 and 7
- Split:
$ 3x^2 + 18x + 7x + 42 $
- Group:
$ (3x^2 + 18x) + (7x + 42) $
- Factor:
$ 3x(x + 6) + 7(x + 6) $
- Binomial:
$ (3x + 7)(x + 6) $

✔ Answer: $ \boxed{(3x + 7)(x + 6)} $

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9. $ 2x^2 - 3x - 20 $

- $ a \cdot c = 2 \cdot (-20) = -40 $
- Numbers: multiply to -40, add to -3 → -8 and 5
- Split:
$ 2x^2 - 8x + 5x - 20 $
- Group:
$ (2x^2 - 8x) + (5x - 20) $
- Factor:
$ 2x(x - 4) + 5(x - 4) $
- Binomial:
$ (2x + 5)(x - 4) $

✔ Answer: $ \boxed{(2x + 5)(x - 4)} $

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10. $ 35x^2 - 11x - 6 $

- $ a \cdot c = 35 \cdot (-6) = -210 $
- Numbers: multiply to -210, add to -11 → -21 and 10
- Split:
$ 35x^2 - 21x + 10x - 6 $
- Group:
$ (35x^2 - 21x) + (10x - 6) $
- Factor:
$ 7x(5x - 3) + 2(5x - 3) $
- Binomial:
$ (7x + 2)(5x - 3) $

✔ Answer: $ \boxed{(7x + 2)(5x - 3)} $

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## ✔ Final Answers Summary:

1. $ \boxed{(2x + 7)(2x - 5)} $
2. $ \boxed{(5x - 3)(x + 8)} $
3. $ \boxed{3(x + 2)^2} $
4. $ \boxed{2(3x - 1)(x - 7)} $
5. $ \boxed{(3x + 4)(x - 9)} $
6. $ \boxed{(2x - 1)(x - 1)} $
7. $ \boxed{2(2x - 1)(5x + 3)} $
8. $ \boxed{(3x + 7)(x + 6)} $
9. $ \boxed{(2x + 5)(x - 4)} $
10. $ \boxed{(7x + 2)(5x - 3)} $

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