Practice sheet for algebra students focusing on factoring polynomials of the form x² + bx + c after removing the greatest common factor.

Let's solve each problem step by step from your algebra worksheet. The goal is to factor completely the given polynomials and solve equations by factoring, focusing on expressions of the form $ x^2 + bx + c $, including those with greatest common factors (GCFs).

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Part 1: Factor Completely

We'll factor each polynomial completely.

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1. $ 2x^2 + 6x + 4 $

- First, factor out the GCF: $ 2 $
- $ = 2(x^2 + 3x + 2) $
- Now factor the trinomial: $ x^2 + 3x + 2 = (x+1)(x+2) $
- ✔ Final answer: $ \boxed{2(x+1)(x+2)} $

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2. $ 4a^2 - 12a + 8 $

- GCF: $ 4 $
- $ = 4(a^2 - 3a + 2) $
- Factor: $ a^2 - 3a + 2 = (a-1)(a-2) $
- ✔ Final answer: $ \boxed{4(a-1)(a-2)} $

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3. $ 10a^2 + 10 - 20 $

Wait — this seems like a typo. It says $ 10a^2 + 10 - 20 $. That simplifies to:
$ 10a^2 - 10 $

But that’s not a quadratic in standard form. Let’s assume it was meant to be:
> $ 10a^2 + 10a - 20 $

(That would make sense for factoring.)

So:
- GCF: $ 10 $
- $ = 10(a^2 + a - 2) $
- Factor: $ a^2 + a - 2 = (a+2)(a-1) $
- ✔ Final answer: $ \boxed{10(a+2)(a-1)} $

If the original was really $ 10a^2 + 10 - 20 $, then:
- $ = 10a^2 - 10 = 10(a^2 - 1) = 10(a-1)(a+1) $
- But that’s likely not intended.

Let’s proceed assuming it was a typo and should be $ 10a^2 + 10a - 20 $ → as above.

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4. $ 7a^2 - 14a - 21 $

- GCF: $ 7 $
- $ = 7(a^2 - 2a - 3) $
- Factor: $ a^2 - 2a - 3 = (a-3)(a+1) $
- ✔ Final answer: $ \boxed{7(a-3)(a+1)} $

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5. $ 3y^2 - 15y + 18 $

- GCF: $ 3 $
- $ = 3(y^2 - 5y + 6) $
- Factor: $ y^2 - 5y + 6 = (y-2)(y-3) $
- ✔ Final answer: $ \boxed{3(y-2)(y-3)} $

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6. $ a^3 - 5a^2 + 4a $

- GCF: $ a $
- $ = a(a^2 - 5a + 4) $
- Factor: $ a^2 - 5a + 4 = (a-1)(a-4) $
- ✔ Final answer: $ \boxed{a(a-1)(a-4)} $

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7. $ x^4 - 15x^3 + 56x^2 $

- GCF: $ x^2 $
- $ = x^2(x^2 - 15x + 56) $
- Factor: $ x^2 - 15x + 56 = (x-7)(x-8) $
- ✔ Final answer: $ \boxed{x^2(x-7)(x-8)} $

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8. $ b^4 - 3b^3 - 10b^2 $

- GCF: $ b^2 $
- $ = b^2(b^2 - 3b - 10) $
- Factor: $ b^2 - 3b - 10 = (b-5)(b+2) $
- ✔ Final answer: $ \boxed{b^2(b-5)(b+2)} $

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9. $ 2a^3 + 8a^2 - 64a $

- GCF: $ 2a $
- $ = 2a(a^2 + 4a - 32) $
- Factor: $ a^2 + 4a - 32 = (a+8)(a-4) $
- ✔ Final answer: $ \boxed{2a(a+8)(a-4)} $

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10. $ 3a^3 - 9a^2 - 54a $

- GCF: $ 3a $
- $ = 3a(a^2 - 3a - 18) $
- Factor: $ a^2 - 3a - 18 = (a-6)(a+3) $
- ✔ Final answer: $ \boxed{3a(a-6)(a+3)} $

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11. $ 9p^2 - 54p + 72 $

- GCF: $ 9 $
- $ = 9(p^2 - 6p + 8) $
- Factor: $ p^2 - 6p + 8 = (p-2)(p-4) $
- ✔ Final answer: $ \boxed{9(p-2)(p-4)} $

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12. $ 4y^3 - 4y^2 - 24y $

- GCF: $ 4y $
- $ = 4y(y^2 - y - 6) $
- Factor: $ y^2 - y - 6 = (y-3)(y+2) $
- ✔ Final answer: $ \boxed{4y(y-3)(y+2)} $

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13. $ 3x^4 - 21x^3 + 10x^2 $

- GCF: $ x^2 $
- $ = x^2(3x^2 - 21x + 10) $
- Now factor $ 3x^2 - 21x + 10 $

Use AC method:
- $ a=3, b=-21, c=10 $
- $ ac = 3×10 = 30 $
- Find two numbers that multiply to 30 and add to -21 → -20 and -1
- Split middle term: $ 3x^2 - 20x - x + 10 $
- Group: $ (3x^2 - 20x) + (-x + 10) $
- $ = x(3x - 20) -1(x - 10) $ → doesn’t work

Try factoring directly:

Try $ (3x - 1)(x - 10) = 3x^2 - 30x - x + 10 = 3x^2 - 31x + 10 $ → no

Try $ (3x - 2)(x - 5) = 3x^2 - 15x - 2x + 10 = 3x^2 - 17x + 10 $ → no

Try $ (3x - 5)(x - 2) = 3x^2 - 6x - 5x + 10 = 3x^2 - 11x + 10 $ → no

Try $ (3x - 10)(x - 1) = 3x^2 - 3x - 10x + 10 = 3x^2 - 13x + 10 $ → no

None seem to work.

Wait: maybe we made an error. Let's double-check.

Is $ 3x^2 - 21x + 10 $ factorable?

Discriminant: $ (-21)^2 - 4(3)(10) = 441 - 120 = 321 $ → not a perfect square → not factorable over integers

So the expression cannot be factored further using integers.

✔ Final answer: $ \boxed{x^2(3x^2 - 21x + 10)} $ ← already factored completely

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14. $ 5x^4 - 10x^3 - 75x^2 $

- GCF: $ 5x^2 $
- $ = 5x^2(x^2 - 2x - 15) $
- Factor: $ x^2 - 2x - 15 = (x-5)(x+3) $
- ✔ Final answer: $ \boxed{5x^2(x-5)(x+3)} $

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Part 2: Solve Each Equation by Factoring

Set each equation to zero and factor.

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15. $ 3x^2 + 15x + 18 = 0 $

- GCF: $ 3 $
- $ = 3(x^2 + 5x + 6) = 0 $
- $ x^2 + 5x + 6 = (x+2)(x+3) $
- So: $ 3(x+2)(x+3) = 0 $
- Solutions: $ x = -2, x = -3 $
- ✔ Final answer: $ \boxed{x = -2, -3} $

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16. $ 2x^2 + 16x + 24 = 0 $

- GCF: $ 2 $
- $ = 2(x^2 + 8x + 12) = 0 $
- $ x^2 + 8x + 12 = (x+6)(x+2) $
- So: $ 2(x+6)(x+2) = 0 $
- Solutions: $ x = -6, x = -2 $
- ✔ Final answer: $ \boxed{x = -6, -2} $

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17. $ 5x^2 - 35x + 60 = 0 $

- GCF: $ 5 $
- $ = 5(x^2 - 7x + 12) = 0 $
- $ x^2 - 7x + 12 = (x-3)(x-4) $
- So: $ 5(x-3)(x-4) = 0 $
- Solutions: $ x = 3, x = 4 $
- ✔ Final answer: $ \boxed{x = 3, 4} $

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18. $ x^3 + 11x^2 - 12x = 0 $

- GCF: $ x $
- $ = x(x^2 + 11x - 12) = 0 $
- Factor: $ x^2 + 11x - 12 = (x+12)(x-1) $
- So: $ x(x+12)(x-1) = 0 $
- Solutions: $ x = 0, x = -12, x = 1 $
- ✔ Final answer: $ \boxed{x = 0, -12, 1} $

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19. $ 2y^2 + 10y = 28 $

First, move all terms to one side:
- $ 2y^2 + 10y - 28 = 0 $
- GCF: $ 2 $
- $ = 2(y^2 + 5y - 14) = 0 $
- Now factor $ y^2 + 5y - 14 $

Check discriminant: $ 25 + 56 = 81 $ → perfect square!

Roots: $ y = \frac{-5 \pm \sqrt{81}}{2} = \frac{-5 \pm 9}{2} $

So:
- $ y = \frac{4}{2} = 2 $
- $ y = \frac{-14}{2} = -7 $

So: $ y^2 + 5y - 14 = (y-2)(y+7) $

Thus: $ 2(y-2)(y+7) = 0 $

Solutions: $ y = 2, y = -7 $

✔ Final answer: $ \boxed{y = 2, -7} $

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20. $ 6y^2 + 36 = 30y $

Move all terms to one side:
- $ 6y^2 - 30y + 36 = 0 $
- GCF: $ 6 $
- $ = 6(y^2 - 5y + 6) = 0 $
- $ y^2 - 5y + 6 = (y-2)(y-3) $
- So: $ 6(y-2)(y-3) = 0 $
- Solutions: $ y = 2, y = 3 $
- ✔ Final answer: $ \boxed{y = 2, 3} $

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✔ Summary of Answers:

#### Factor Completely
1. $ 2(x+1)(x+2) $
2. $ 4(a-1)(a-2) $
3. $ 10(a+2)(a-1) $ *(assuming typo)*
4. $ 7(a-3)(a+1) $
5. $ 3(y-2)(y-3) $
6. $ a(a-1)(a-4) $
7. $ x^2(x-7)(x-8) $
8. $ b^2(b-5)(b+2) $
9. $ 2a(a+8)(a-4) $
10. $ 3a(a-6)(a+3) $
11. $ 9(p-2)(p-4) $
12. $ 4y(y-3)(y+2) $
13. $ x^2(3x^2 - 21x + 10) $ *(not factorable further)*
14. $ 5x^2(x-5)(x+3) $

#### Solve by Factoring
15. $ x = -2, -3 $
16. $ x = -6, -2 $
17. $ x = 3, 4 $
18. $ x = 0, -12, 1 $
19. $ y = 2, -7 $
20. $ y = 2, 3 $

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Let me know if you'd like these checked against the original image or need explanations for any specific steps!