Factoring Trinomials worksheet with 22 algebra problems for practice.

Here are the factored forms for each trinomial. I have worked through them step-by-step to ensure accuracy.

1) $x^2 - 121$
This is a difference of squares ($a^2 - b^2$).
$\sqrt{x^2} = x$ and $\sqrt{121} = 11$.
So, it factors into $(x - 11)(x + 11)$.

2) $x^2 - 7x - 18$
We need two numbers that multiply to $-18$ and add to $-7$.
The numbers are $-9$ and $+2$.
$(-9) \times 2 = -18$
$(-9) + 2 = -7$
Answer: $(x - 9)(x + 2)$

3) $x^2 + 5x - 36$
We need two numbers that multiply to $-36$ and add to $+5$.
The numbers are $+9$ and $-4$.
$9 \times (-4) = -36$
$9 + (-4) = 5$
Answer: $(x + 9)(x - 4)$

4) $x^2 - 6x - 16$
We need two numbers that multiply to $-16$ and add to $-6$.
The numbers are $-8$ and $+2$.
$(-8) \times 2 = -16$
$(-8) + 2 = -6$
Answer: $(x - 8)(x + 2)$

5) $x^2 + 13x + 36$
We need two numbers that multiply to $+36$ and add to $+13$.
The numbers are $+9$ and $+4$.
$9 \times 4 = 36$
$9 + 4 = 13$
Answer: $(x + 9)(x + 4)$

6) $-12x^2 - 26x - 12$
First, factor out the greatest common factor, which is $-2$.
$-2(6x^2 + 13x + 6)$
Now factor the inside part: $6x^2 + 13x + 6$.
We need numbers that multiply to $6 \times 6 = 36$ and add to $13$. Those are $9$ and $4$.
Split the middle term: $6x^2 + 9x + 4x + 6$.
Group: $3x(2x + 3) + 2(2x + 3)$.
Inside factors to $(3x + 2)(2x + 3)$.
Don't forget the $-2$ from the start.
Answer: $-2(3x + 2)(2x + 3)$

7) $x^2 - 6x + 5$
We need two numbers that multiply to $+5$ and add to $-6$.
The numbers are $-5$ and $-1$.
$(-5) \times (-1) = 5$
$(-5) + (-1) = -6$
Answer: $(x - 5)(x - 1)$

8) $x^2 + 10x + 25$
This is a perfect square trinomial.
We need two numbers that multiply to $25$ and add to $10$.
The numbers are $5$ and $5$.
Answer: $(x + 5)(x + 5)$ or $(x + 5)^2$

9) $x^2 - 1x - 20$
We need two numbers that multiply to $-20$ and add to $-1$.
The numbers are $-5$ and $+4$.
$(-5) \times 4 = -20$
$(-5) + 4 = -1$
Answer: $(x - 5)(x + 4)$

10) $x^2 - 6x + 5$
Same as number 7.
Numbers are $-5$ and $-1$.
Answer: $(x - 5)(x - 1)$

11) $x^2 + 4x - 45$
We need two numbers that multiply to $-45$ and add to $+4$.
The numbers are $+9$ and $-5$.
$9 \times (-5) = -45$
$9 + (-5) = 4$
Answer: $(x + 9)(x - 5)$

12) $x^2 - 5x - 14$
We need two numbers that multiply to $-14$ and add to $-5$.
The numbers are $-7$ and $+2$.
$(-7) \times 2 = -14$
$(-7) + 2 = -5$
Answer: $(x - 7)(x + 2)$

13) $2x^2 + 3x - 5$
Multiply $a \cdot c$: $2 \cdot (-5) = -10$.
Find numbers that multiply to $-10$ and add to $b$ ($3$).
The numbers are $+5$ and $-2$.
Rewrite middle term: $2x^2 - 2x + 5x - 5$.
Factor by grouping:
$2x(x - 1) + 5(x - 1)$.
Answer: $(2x + 5)(x - 1)$

14) $x^2 + 8x + 15$
We need two numbers that multiply to $15$ and add to $8$.
The numbers are $5$ and $3$.
Answer: $(x + 5)(x + 3)$

15) $x^2 - 64$
Difference of squares.
$\sqrt{x^2} = x$, $\sqrt{64} = 8$.
Answer: $(x - 8)(x + 8)$

16) $x^2 - 2x - 15$
We need two numbers that multiply to $-15$ and add to $-2$.
The numbers are $-5$ and $+3$.
$(-5) \times 3 = -15$
$(-5) + 3 = -2$
Answer: $(x - 5)(x + 3)$

17) $x^2 - 2x - 15$
Same as number 16.
Answer: $(x - 5)(x + 3)$

18) $-10x^2 - 24x - 8$
Factor out GCF $-2$:
$-2(5x^2 + 12x + 4)$
Factor inside: $5x^2 + 12x + 4$.
Multiply $a \cdot c$: $5 \cdot 4 = 20$.
Find numbers multiplying to $20$ and adding to $12$. They are $10$ and $2$.
Rewrite: $5x^2 + 10x + 2x + 4$.
Group: $5x(x + 2) + 2(x + 2)$.
Inside factors to $(5x + 2)(x + 2)$.
Include the $-2$.
Answer: $-2(5x + 2)(x + 2)$

19) $x^2 - 9x + 18$
We need two numbers that multiply to $18$ and add to $-9$.
The numbers are $-6$ and $-3$.
$(-6) \times (-3) = 18$
$(-6) + (-3) = -9$
Answer: $(x - 6)(x - 3)$

20) $x^2 - 4x - 5$
We need two numbers that multiply to $-5$ and add to $-4$.
The numbers are $-5$ and $+1$.
$(-5) \times 1 = -5$
$(-5) + 1 = -4$
Answer: $(x - 5)(x + 1)$

21) $x^2 - 11x + 18$
We need two numbers that multiply to $18$ and add to $-11$.
The numbers are $-9$ and $-2$.
$(-9) \times (-2) = 18$
$(-9) + (-2) = -11$
Answer: $(x - 9)(x - 2)$

22) $x^2 - 9x + 14$
We need two numbers that multiply to $14$ and add to $-9$.
The numbers are $-7$ and $-2$.
$(-7) \times (-2) = 14$
$(-7) + (-2) = -9$
Answer: $(x - 7)(x - 2)$

Final Answer:
1) $(x - 11)(x + 11)$
2) $(x - 9)(x + 2)$
3) $(x + 9)(x - 4)$
4) $(x - 8)(x + 2)$
5) $(x + 9)(x + 4)$
6) $-2(3x + 2)(2x + 3)$
7) $(x - 5)(x - 1)$
8) $(x + 5)(x + 5)$
9) $(x - 5)(x + 4)$
10) $(x - 5)(x - 1)$
11) $(x + 9)(x - 5)$
12) $(x - 7)(x + 2)$
13) $(2x + 5)(x - 1)$
14) $(x + 5)(x + 3)$
15) $(x - 8)(x + 8)$
16) $(x - 5)(x + 3)$
17) $(x - 5)(x + 3)$
18) $-2(5x + 2)(x + 2)$
19) $(x - 6)(x - 3)$
20) $(x - 5)(x + 1)$
21) $(x - 9)(x - 2)$
22) $(x - 7)(x - 2)$