Factoring Trinomials Worksheets - Math Monks - Free Printable

Let's solve each trinomial factoring problem step by step. We'll use factoring by grouping, AC method, or trial and error depending on the structure of the trinomial.

---

1. $ 2p^2 + 6p - 108 $

Step 1: Factor out the GCF (greatest common factor).
All terms are divisible by 2:
$$
= 2(p^2 + 3p - 54)
$$

Step 2: Factor $ p^2 + 3p - 54 $.
We need two numbers that multiply to $-54$ and add to $3$.
Try: $9$ and $-6$: $9 \times (-6) = -54$, $9 + (-6) = 3$ ✔
So:
$$
p^2 + 3p - 54 = (p + 9)(p - 6)
$$

Final Answer:
$$
\boxed{2(p + 9)(p - 6)}
$$

---

2. $ 7b^2 + 53b + 28 $

Use the AC method:
$ A = 7 $, $ B = 53 $, $ C = 28 $
$ AC = 7 \times 28 = 196 $
Find two numbers that multiply to 196 and add to 53.

Try: $49$ and $4$: $49 \times 4 = 196$, $49 + 4 = 53$ ✔

Now split the middle term:
$$
7b^2 + 49b + 4b + 28
$$

Group:
$$
(7b^2 + 49b) + (4b + 28) = 7b(b + 7) + 4(b + 7)
$$

Factor:
$$
(7b + 4)(b + 7)
$$

Final Answer:
$$
\boxed{(7b + 4)(b + 7)}
$$

---

3. $ 6x^2 + 7x - 49 $

$ A = 6 $, $ B = 7 $, $ C = -49 $
$ AC = 6 \times (-49) = -294 $
Need two numbers that multiply to $-294$ and add to $7$

Try: $21$ and $-14$: $21 \times (-14) = -294$, $21 + (-14) = 7$ ✔

Split:
$$
6x^2 + 21x - 14x - 49
$$

Group:
$$
(6x^2 + 21x) + (-14x - 49) = 3x(2x + 7) -7(2x + 7)
$$

Wait — check:
$3x(2x + 7) = 6x^2 + 21x$ ✔
$-7(2x + 7) = -14x - 49$ ✔

So:
$$
(3x - 7)(2x + 7)
$$

Final Answer:
$$
\boxed{(3x - 7)(2x + 7)}
$$

---

4. $ 3p^2 - 3p - 36 $

Factor out GCF: all terms divisible by 3
$$
= 3(p^2 - p - 12)
$$

Now factor $ p^2 - p - 12 $:
Need two numbers that multiply to $-12$, add to $-1$:
Try: $-4$ and $3$: $-4 \times 3 = -12$, $-4 + 3 = -1$ ✔
So:
$$
p^2 - p - 12 = (p - 4)(p + 3)
$$

Final Answer:
$$
\boxed{3(p - 4)(p + 3)}
$$

---

5. $ 5q^2 + 19q + 12 $

$ A = 5 $, $ B = 19 $, $ C = 12 $
$ AC = 5 \times 12 = 60 $
Need two numbers that multiply to 60 and add to 19:
Try: $15$ and $4$: $15 \times 4 = 60$, $15 + 4 = 19$ ✔

Split:
$$
5q^2 + 15q + 4q + 12
$$

Group:
$$
(5q^2 + 15q) + (4q + 12) = 5q(q + 3) + 4(q + 3)
$$

Factor:
$$
(5q + 4)(q + 3)
$$

Final Answer:
$$
\boxed{(5q + 4)(q + 3)}
$$

---

6. $ x^2 + 12x - 13 $

Need two numbers that multiply to $-13$, add to $12$:
Try: $13$ and $-1$: $13 \times (-1) = -13$, $13 + (-1) = 12$ ✔

So:
$$
x^2 + 12x - 13 = (x + 13)(x - 1)
$$

Final Answer:
$$
\boxed{(x + 13)(x - 1)}
$$

---

7. $ 3x^2 - 12x - 135 $

Factor out GCF: all divisible by 3
$$
= 3(x^2 - 4x - 45)
$$

Now factor $ x^2 - 4x - 45 $:
Need two numbers that multiply to $-45$, add to $-4$:
Try: $-9$ and $5$: $-9 \times 5 = -45$, $-9 + 5 = -4$ ✔

So:
$$
x^2 - 4x - 45 = (x - 9)(x + 5)
$$

Final Answer:
$$
\boxed{3(x - 9)(x + 5)}
$$

---

8. $ 5m^2 - 11m - 12 $

$ A = 5 $, $ B = -11 $, $ C = -12 $
$ AC = 5 \times (-12) = -60 $
Need two numbers that multiply to $-60$, add to $-11$:
Try: $-15$ and $4$: $-15 \times 4 = -60$, $-15 + 4 = -11$ ✔

Split:
$$
5m^2 - 15m + 4m - 12
$$

Group:
$$
(5m^2 - 15m) + (4m - 12) = 5m(m - 3) + 4(m - 3)
$$

Factor:
$$
(5m + 4)(m - 3)
$$

Final Answer:
$$
\boxed{(5m + 4)(m - 3)}
$$

---

9. $ 56p^2 + 64p + 8 $

Factor out GCF: all divisible by 8
$$
= 8(7p^2 + 8p + 1)
$$

Now factor $ 7p^2 + 8p + 1 $:
$ AC = 7 \times 1 = 7 $
Need two numbers that multiply to 7, add to 8:
Try: $7$ and $1$: $7 \times 1 = 7$, $7 + 1 = 8$ ✔

Split:
$$
7p^2 + 7p + p + 1
$$

Group:
$$
(7p^2 + 7p) + (p + 1) = 7p(p + 1) + 1(p + 1)
$$

Factor:
$$
(7p + 1)(p + 1)
$$

Final Answer:
$$
\boxed{8(7p + 1)(p + 1)}
$$

---

10. $ 25s^2 + 30s - 16 $

$ A = 25 $, $ B = 30 $, $ C = -16 $
$ AC = 25 \times (-16) = -400 $
Need two numbers that multiply to $-400$, add to $30$:
Try: $40$ and $-10$: $40 \times (-10) = -400$, $40 + (-10) = 30$ ✔

Split:
$$
25s^2 + 40s - 10s - 16
$$

Group:
$$
(25s^2 + 40s) + (-10s - 16) = 5s(5s + 8) -2(5s + 8)
$$

Factor:
$$
(5s - 2)(5s + 8)
$$

Final Answer:
$$
\boxed{(5s - 2)(5s + 8)}
$$

---

✔ Final Answers Summary:

| Problem | Factored Form |
|--------|----------------|
| 1 | $ 2(p + 9)(p - 6) $ |
| 2 | $ (7b + 4)(b + 7) $ |
| 3 | $ (3x - 7)(2x + 7) $ |
| 4 | $ 3(p - 4)(p + 3) $ |
| 5 | $ (5q + 4)(q + 3) $ |
| 6 | $ (x + 13)(x - 1) $ |
| 7 | $ 3(x - 9)(x + 5) $ |
| 8 | $ (5m + 4)(m - 3) $ |
| 9 | $ 8(7p + 1)(p + 1) $ |
| 10 | $ (5s - 2)(5s + 8) $ |

Let me know if you'd like a PDF version or explanation for any specific one!