I’ll solve each trinomial by factoring completely. Since all have a > 1, I’ll use the “ac method” or trial and error with grouping.
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1. 4x² + 4x – 35
Multiply a·c = 4·(-35) = -140
Find two numbers that multiply to -140 and add to 4 → 14 and -10
Rewrite middle term: 4x² + 14x – 10x – 35
Group: (4x² + 14x) + (-10x – 35)
Factor each group: 2x(2x + 7) –5(2x + 7)
Final:
(2x – 5)(2x + 7)
✔ Check: (2x–5)(2x+7) = 4x² +14x –10x –35 = 4x² +4x –35 ✔️
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2. 5x² + 37x – 24
a·c = 5·(-24) = -120
Numbers that multiply to -120, add to 37 → 40 and -3
Rewrite: 5x² + 40x – 3x – 24
Group: (5x² + 40x) + (-3x – 24)
Factor: 5x(x + 8) –3(x + 8)
Final:
(5x – 3)(x + 8)
✔ Check: (5x–3)(x+8) = 5x² +40x –3x –24 = 5x² +37x –24 ✔️
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3. 3x² + 12x + 12
First, factor out GCF: 3(x² + 4x + 4)
Now factor inside: x² + 4x + 4 = (x + 2)²
Final:
3(x + 2)²
✔ Check: 3(x+2)² = 3(x²+4x+4) = 3x²+12x+12 ✔️
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4. 6x² – 44x + 14
GCF first: 2(3x² – 22x + 7)
Now factor 3x² –22x +7
a·c = 3·7=21; need factors of 21 that add to -22 → -21 and -1
Rewrite: 3x² –21x –x +7
Group: (3x² –21x) + (-x +7) → 3x(x–7) –1(x–7)
So: (3x–1)(x–7)
Final:
2(3x – 1)(x – 7)
✔ Check: 2(3x–1)(x–7) = 2[3x²–21x–x+7] = 2[3x²–22x+7] = 6x²–44x+14 ✔️
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5. 3x² – 23x – 36
a·c = 3·(-36) = -108
Need factors of -108 that add to -23 → -27 and 4
Rewrite: 3x² –27x +4x –36
Group: (3x² –27x) + (4x –36) → 3x(x–9) +4(x–9)
Final:
(3x + 4)(x – 9)
✔ Check: (3x+4)(x–9) = 3x²–27x+4x–36 = 3x²–23x–36 ✔️
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6. 2x² – 3x + 1
a·c = 2·1 = 2
Factors of 2 that add to -3 → -2 and -1
Rewrite: 2x² –2x –x +1
Group: (2x² –2x) + (-x +1) → 2x(x–1) –1(x–1)
Final:
(2x – 1)(x – 1)
✔ Check: (2x–1)(x–1) = 2x²–2x–x+1 = 2x²–3x+1 ✔️
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7. 20x² + 2x – 6
GCF first: 2(10x² + x – 3)
Factor 10x² + x –3
a·c = 10·(-3) = -30
Factors of -30 that add to 1 → 6 and -5
Rewrite: 10x² +6x –5x –3
Group: (10x² +6x) + (-5x –3) → 2x(5x+3) –1(5x+3)
So: (2x–1)(5x+3)
Final:
2(2x – 1)(5x + 3)
✔ Check: 2(2x–1)(5x+3) = 2[10x²+6x–5x–3] = 2[10x²+x–3] = 20x²+2x–6 ✔️
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8. 3x² + 25x + 42
a·c = 3·42 = 126
Factors of 126 that add to 25 → 18 and 7
Rewrite: 3x² +18x +7x +42
Group: (3x² +18x) + (7x +42) → 3x(x+6) +7(x+6)
Final:
(3x + 7)(x + 6)
✔ Check: (3x+7)(x+6) = 3x²+18x+7x+42 = 3x²+25x+42 ✔️
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9. 2x² – 3x – 20
a·c = 2·(-20) = -40
Factors of -40 that add to -3 → -8 and 5
Rewrite: 2x² –8x +5x –20
Group: (2x² –8x) + (5x –20) → 2x(x–4) +5(x–4)
Final:
(2x + 5)(x – 4)
✔ Check: (2x+5)(x–4) = 2x²–8x+5x–20 = 2x²–3x–20 ✔️
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10. 35x² – 11x – 6
a·c = 35·(-6) = -210
Factors of -210 that add to -11 → -21 and 10
Rewrite: 35x² –21x +10x –6
Group: (35x² –21x) + (10x –6) → 7x(5x–3) +2(5x–3)
Final:
(7x + 2)(5x – 3)
✔ Check: (7x+2)(5x–3) = 35x²–21x+10x–6 = 35x²–11x–6 ✔️
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Final Answer:
1. (2x – 5)(2x + 7)
2. (5x – 3)(x + 8)
3. 3(x + 2)²
4. 2(3x – 1)(x – 7)
5. (3x + 4)(x – 9)
6. (2x – 1)(x – 1)
7. 2(2x – 1)(5x + 3)
8. (3x + 7)(x + 6)
9. (2x + 5)(x – 4)
10. (7x + 2)(5x – 3)
Parent Tip: Review the logic above to help your child master the concept of factoring trinomials worksheet with answer key.
