Factoring Trinomials worksheets - Free Printable

Here is the complete solution to Factoring Trinomials worksheet. Each trinomial is factored using standard techniques: finding two numbers that multiply to the constant term (or ac for non-monic) and add to the middle coefficient, then grouping or using the AC method where needed.

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✔ Solutions:

1) \( x^2 + 11x + 18 = \boxed{(x + 2)(x + 9)} \)
→ 2×9=18, 2+9=11

2) \( x^2 + 2x - 99 = \boxed{(x + 11)(x - 9)} \)
→ 11×(-9)= -99, 11 + (-9) = 2

3) \( x^2 - 2x - 35 = \boxed{(x - 7)(x + 5)} \)
→ -7×5 = -35, -7 + 5 = -2

4) \( -2x^2 - 11x - 9 = \boxed{-(2x + 9)(x + 1)} \)
→ Factor out -1: -(2x² + 11x + 9). Then factor: 2×9=18; 9+2=11 → (2x+9)(x+1)

5) \( x^2 - x - 20 = \boxed{(x - 5)(x + 4)} \)
→ -5×4 = -20, -5+4 = -1

6) \( x^2 - 64 = \boxed{(x - 8)(x + 8)} \)
→ Difference of squares

7) \( x^2 - 6x + 8 = \boxed{(x - 2)(x - 4)} \)
→ -2×-4=8, -2+-4=-6

8) \( x^2 + 9x + 14 = \boxed{(x + 2)(x + 7)} \)
→ 2×7=14, 2+7=9

9) \( 4x^2 + 3x - 7 = \boxed{(4x + 7)(x - 1)} \)
→ AC=4×(-7)= -28; find factors of -28 that add to 3: 7 and -4 → split: 4x² +7x -4x -7 → group: x(4x+7)-1(4x+7)

10) \( x^2 - 12x + 20 = \boxed{(x - 2)(x - 10)} \)
→ -2×-10=20, -2+-10=-12

11) \( x^2 - 5x - 6 = \boxed{(x - 6)(x + 1)} \)
→ -6×1=-6, -6+1=-5

12) \( x^2 - x - 90 = \boxed{(x - 10)(x + 9)} \)
→ -10×9=-90, -10+9=-1

13) \( -7x^2 - 9x - 2 = \boxed{-(7x + 2)(x + 1)} \)
→ Factor out -1: -(7x² + 9x + 2). AC=14; factors 7&2 → (7x+2)(x+1)

14) \( x^2 - 2x - 35 = \boxed{(x - 7)(x + 5)} \)
→ Same as #3

15) \( x^2 + x - 30 = \boxed{(x + 6)(x - 5)} \)
→ 6×(-5)=-30, 6-5=1

16) \( x^2 - 2x - 99 = \boxed{(x - 11)(x + 9)} \)
→ -11×9=-99, -11+9=-2

17) \( x^2 + 16x + 63 = \boxed{(x + 7)(x + 9)} \)
→ 7×9=63, 7+9=16

18) \( x^2 - 18x + 80 = \boxed{(x - 8)(x - 10)} \)
→ -8×-10=80, -8+-10=-18

19) \( x^2 + 15x + 56 = \boxed{(x + 7)(x + 8)} \)
→ 7×8=56, 7+8=15

20) \( x^2 - 8x + 7 = \boxed{(x - 1)(x - 7)} \)
→ -1×-7=7, -1+-7=-8

21) \( x^2 + 17x + 72 = \boxed{(x + 8)(x + 9)} \)
→ 8×9=72, 8+9=17

22) \( x^2 + 19x + 88 = \boxed{(x + 8)(x + 11)} \)
→ 8×11=88, 8+11=19

23) \( x^2 + 3x - 4 = \boxed{(x + 4)(x - 1)} \)
→ 4×(-1)=-4, 4-1=3

24) \( x^2 - x - 30 = \boxed{(x - 6)(x + 5)} \)
→ -6×5=-30, -6+5=-1

25) \( x^2 + 16x + 63 = \boxed{(x + 7)(x + 9)} \)
→ Same as #17

26) \( x^2 + 9x + 14 = \boxed{(x + 2)(x + 7)} \)
→ Same as #8

27) \( 15x^2 + 10x - 5 = \boxed{5(3x - 1)(x + 1)} \)
→ Factor out GCF 5: 5(3x² + 2x -1). Then factor: 3×(-1)=-3; factors 3,-1 → 3x²+3x-x-1 → 3x(x+1)-1(x+1) → (3x-1)(x+1)

28) \( -12x^2 - 26x - 12 = \boxed{-2(2x + 3)(3x + 2)} \)
→ Factor out -2: -2(6x² + 13x + 6). AC=36; factors 9,4 → 6x²+9x+4x+6 → 3x(2x+3)+2(2x+3) → (3x+2)(2x+3)

29) \( x^2 + 7x + 10 = \boxed{(x + 2)(x + 5)} \)
→ 2×5=10, 2+5=7

30) \( x^2 + 11x + 28 = \boxed{(x + 4)(x + 7)} \)
→ 4×7=28, 4+7=11

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📌 Key Factoring Tips:

- For \( x^2 + bx + c \): Find two numbers that multiply to c and add to b.
- For \( ax^2 + bx + c \) (a ≠ 1): Use AC method — multiply a×c, find factors that add to b, split middle term, then factor by grouping.
- Always check for GCF first!
- Difference of squares: \( a^2 - b^2 = (a - b)(a + b) \)
- If leading coefficient is negative, factor out -1 first if helpful.

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✔ All problems solved with explanations. Let me know if you want step-by-step breakdowns for any specific one!