Factoring trinomials practice worksheet with six quadratic expressions to factor.

Here are the factored forms of each trinomial, solved step by step using the “ac method” or trial-and-error with grouping — perfect for school-level algebra.

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1. 2x² + 13x + 21

Multiply a·c = 2·21 = 42
Find two numbers that multiply to 42 and add to 13 → 6 and 7
Rewrite middle term: 2x² + 6x + 7x + 21
Group: (2x² + 6x) + (7x + 21)
Factor each group: 2x(x + 3) + 7(x + 3)
→ (2x + 7)(x + 3)

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2. 5x² + 54x + 40

a·c = 5·40 = 200
Numbers that multiply to 200 and add to 54? Try 4 and 50 → yes!
Rewrite: 5x² + 4x + 50x + 40
Group: (5x² + 4x) + (50x + 40)
Factor: x(5x + 4) + 10(5x + 4)
→ (x + 10)(5x + 4)

Wait — check: (x+10)(5x+4) = 5x² + 4x + 50x + 40 = 5x² + 54x + 40 ✔

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3. 3x² + 11x + 10

a·c = 3·10 = 30
Numbers: 5 and 6 → 5×6=30, 5+6=11
Rewrite: 3x² + 5x + 6x + 10
Group: (3x² + 5x) + (6x + 10)
Factor: x(3x + 5) + 2(3x + 5)
→ (x + 2)(3x + 5)

Check: (x+2)(3x+5) = 3x² + 5x + 6x + 10 = 3x² + 11x + 10 ✔

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4. 15x² + 19x + 6

a·c = 15·6 = 90
Numbers that multiply to 90 and add to 19 → 9 and 10
Rewrite: 15x² + 9x + 10x + 6
Group: (15x² + 9x) + (10x + 6)
Factor: 3x(5x + 3) + 2(5x + 3)
→ (3x + 2)(5x + 3)

Check: (3x+2)(5x+3) = 15x² + 9x + 10x + 6 = 15x² + 19x + 6 ✔

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5. 2x² + 11x + 12

a·c = 2·12 = 24
Numbers: 8 and 3 → 8×3=24, 8+3=11
Rewrite: 2x² + 8x + 3x + 12
Group: (2x² + 8x) + (3x + 12)
Factor: 2x(x + 4) + 3(x + 4)
→ (2x + 3)(x + 4)

Check: (2x+3)(x+4) = 2x² + 8x + 3x + 12 = 2x² + 11x + 12 ✔

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6. 8x² + 55x + 42

a·c = 8·42 = 336
Need two numbers that multiply to 336 and add to 55.

Try factors of 336:
- 7 × 48 = 336 → 7+48=55 ✔

Rewrite: 8x² + 7x + 48x + 42
Group: (8x² + 7x) + (48x + 42)
Factor: x(8x + 7) + 6(8x + 7)
→ (x + 6)(8x + 7)

Check: (x+6)(8x+7) = 8x² + 7x + 48x + 42 = 8x² + 55x + 42 ✔

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7. 5x² + 11x + 2

a·c = 5·2 = 10
Numbers: 1 and 10 → 1×10=10, 1+10=11
Rewrite: 5x² + x + 10x + 2
Group: (5x² + x) + (10x + 2)
Factor: x(5x + 1) + 2(5x + 1)
→ (x + 2)(5x + 1)

Check: (x+2)(5x+1) = 5x² + x + 10x + 2 = 5x² + 11x + 2 ✔

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8. 9x² + 28x + 20

a·c = 9·20 = 180
Numbers that multiply to 180 and add to 28 → 10 and 18
Rewrite: 9x² + 10x + 18x + 20
Group: (9x² + 10x) + (18x + 20)
Factor: x(9x + 10) + 2(9x + 10)
→ (x + 2)(9x + 10)

Check: (x+2)(9x+10) = 9x² + 10x + 18x + 20 = 9x² + 28x + 20 ✔

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9. 5x² + 16x + 12

a·c = 5·12 = 60
Numbers: 6 and 10 → 6×10=60, 6+10=16
Rewrite: 5x² + 6x + 10x + 12
Group: (5x² + 6x) + (10x + 12)
Factor: x(5x + 6) + 2(5x + 6)
→ (x + 2)(5x + 6)

Check: (x+2)(5x+6) = 5x² + 6x + 10x + 12 = 5x² + 16x + 12 ✔

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10. 6x² + 31x + 28

a·c = 6·28 = 168
Numbers that multiply to 168 and add to 31 → try 7 and 24 → 7×24=168, 7+24=31 ✔

Rewrite: 6x² + 7x + 24x + 28
Group: (6x² + 7x) + (24x + 28)
Factor: x(6x + 7) + 4(6x + 7)
→ (x + 4)(6x + 7)

Check: (x+4)(6x+7) = 6x² + 7x + 24x + 28 = 6x² + 31x + 28 ✔

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Final Answer:

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