This Algebra 2 worksheet provides a comprehensive mixed review of factoring techniques, ranging from simple trinomials to complex polynomials.

Let's solve each of these factoring problems completely, using various factoring methods such as:

- Factoring out the GCF (Greatest Common Factor)
- Factoring trinomials (using AC method or trial and error)
- Difference of squares
- Grouping
- Factoring cubic polynomials

We’ll go step-by-step for each problem.

---

1) $ b^2 - 19b + 90 $

We need two numbers that multiply to 90 and add to -19.
Try: $-9$ and $-10$: $(-9)(-10)=90$, $-9 + (-10) = -19$

✔ So:
$$
b^2 - 19b + 90 = (b - 9)(b - 10)
$$

---

2) $ b^2 + 2b - 24 $

Two numbers that multiply to -24 and add to 2:
Try: $6$ and $-4$: $6 \times (-4) = -24$, $6 + (-4) = 2$

✔
$$
(b + 6)(b - 4)
$$

---

3) $ x^2 + 13x + 30 $

Multiply to 30, add to 13: $10$ and $3$: $10+3=13$, $10×3=30$

✔
$$
(x + 10)(x + 3)
$$

---

4) $ p^2 - 3p - 70 $

Need two numbers that multiply to -70 and add to -3:
Try: $-10$ and $7$: $-10 × 7 = -70$, $-10 + 7 = -3$

✔
$$
(p - 10)(p + 7)
$$

---

5) $ 4k^2 + 20k - 96 $

First, factor out GCF: 4
$$
4(k^2 + 5k - 24)
$$

Now factor $k^2 + 5k - 24$:
Numbers: $8$ and $-3$: $8 × (-3) = -24$, $8 + (-3) = 5$

✔
$$
4(k + 8)(k - 3)
$$

---

6) $ 4n^2 - 8n - 252 $

Factor out GCF: 4
$$
4(n^2 - 2n - 63)
$$

Factor $n^2 - 2n - 63$:
Numbers: $-9$ and $7$: $-9×7 = -63$, $-9+7 = -2$

✔
$$
4(n - 9)(n + 7)
$$

---

7) $ 6n^2 + 42n - 48 $

GCF: 6
$$
6(n^2 + 7n - 8)
$$

Factor $n^2 + 7n - 8$:
Numbers: $8$ and $-1$: $8×(-1)=-8$, $8+(-1)=7$

✔
$$
6(n + 8)(n - 1)
$$

---

8) $ 4x^2 + 12x - 40 $

GCF: 4
$$
4(x^2 + 3x - 10)
$$

Factor $x^2 + 3x - 10$:
Numbers: $5$ and $-2$: $5×(-2)=-10$, $5+(-2)=3$

✔
$$
4(x + 5)(x - 2)
$$

---

9) $ 6x^2 - 45x + 21 $

GCF: 3
$$
3(2x^2 - 15x + 7)
$$

Now factor $2x^2 - 15x + 7$:
Use AC method: $A=2$, $C=7$, $AC=14$
Find two numbers that multiply to 14 and add to -15: $-14$ and $-1$

Rewrite middle term:
$$
2x^2 -14x -x +7 = 2x(x - 7) -1(x - 7) = (2x - 1)(x - 7)
$$

✔
$$
3(2x - 1)(x - 7)
$$

---

10) $ 15b^2 + 57b - 90 $

GCF: 3
$$
3(5b^2 + 19b - 30)
$$

Now factor $5b^2 + 19b - 30$:
AC method: $5×(-30) = -150$
Find two numbers: $25$ and $-6$: $25×(-6) = -150$, $25 + (-6) = 19$

Split:
$$
5b^2 + 25b - 6b - 30 = 5b(b + 5) -6(b + 5) = (5b - 6)(b + 5)
$$

✔
$$
3(5b - 6)(b + 5)
$$

---

11) $ 42a^2 - 354a - 216 $

GCF: 6
$$
6(7a^2 - 59a - 36)
$$

Now factor $7a^2 - 59a - 36$:
AC = $7×(-36) = -252$
Find two numbers: $-63$ and $4$: $-63×4 = -252$, $-63+4 = -59$

Split:
$$
7a^2 -63a +4a -36 = 7a(a - 9) +4(a - 9) = (7a + 4)(a - 9)
$$

✔
$$
6(7a + 4)(a - 9)
$$

---

12) $ 21a^2 + 33a - 90 $

GCF: 3
$$
3(7a^2 + 11a - 30)
$$

AC = $7×(-30) = -210$
Find two numbers: $21$ and $-10$: $21×(-10) = -210$, $21 + (-10) = 11$

Split:
$$
7a^2 +21a -10a -30 = 7a(a + 3) -10(a + 3) = (7a - 10)(a + 3)
$$

✔
$$
3(7a - 10)(a + 3)
$$

---

13) $ 15r^2 + 21r + 6 $

GCF: 3
$$
3(5r^2 + 7r + 2)
$$

Factor $5r^2 + 7r + 2$:
AC = $5×2 = 10$, find numbers: $5$ and $2$: $5+2=7$

Split:
$$
5r^2 +5r +2r +2 = 5r(r + 1) +2(r + 1) = (5r + 2)(r + 1)
$$

✔
$$
3(5r + 2)(r + 1)
$$

---

14) $ 10a^2 - 54a - 112 $

GCF: 2
$$
2(5a^2 - 27a - 56)
$$

AC = $5×(-56) = -280$
Find: $-35$ and $8$: $-35×8 = -280$, $-35+8 = -27$

Split:
$$
5a^2 -35a +8a -56 = 5a(a - 7) +8(a - 7) = (5a + 8)(a - 7)
$$

✔
$$
2(5a + 8)(a - 7)
$$

---

15) $ 9r^2 + 51r + 30 $

GCF: 3
$$
3(3r^2 + 17r + 10)
$$

AC = $3×10 = 30$, numbers: $15$ and $2$: $15+2=17$

Split:
$$
3r^2 +15r +2r +10 = 3r(r + 5) +2(r + 5) = (3r + 2)(r + 5)
$$

✔
$$
3(3r + 2)(r + 5)
$$

---

16) $ 9k^2 + 3k - 12 $

GCF: 3
$$
3(3k^2 + k - 4)
$$

AC = $3×(-4) = -12$, numbers: $4$ and $-3$: $4 + (-3) = 1$

Split:
$$
3k^2 +4k -3k -4 = k(3k + 4) -1(3k + 4) = (k - 1)(3k + 4)
$$

✔
$$
3(k - 1)(3k + 4)
$$

---

17) $ 15n^2 - 110n + 200 $

GCF: 5
$$
5(3n^2 - 22n + 40)
$$

AC = $3×40 = 120$, numbers: $-12$ and $-10$: $-12×(-10)=120$, $-12 + (-10) = -22$

Split:
$$
3n^2 -12n -10n +40 = 3n(n - 4) -10(n - 4) = (3n - 10)(n - 4)
$$

✔
$$
5(3n - 10)(n - 4)
$$

---

18) $ 6p^2 + 33p - 18 $

GCF: 3
$$
3(2p^2 + 11p - 6)
$$

AC = $2×(-6) = -12$, numbers: $12$ and $-1$: $12 + (-1) = 11$

Split:
$$
2p^2 +12p -p -6 = 2p(p + 6) -1(p + 6) = (2p - 1)(p + 6)
$$

✔
$$
3(2p - 1)(p + 6)
$$

---

19) $ -30n^2 + 288n - 384 $

GCF: -6 (to make leading coefficient positive)
$$
-6(5n^2 - 48n + 64)
$$

Now factor $5n^2 - 48n + 64$:
AC = $5×64 = 320$, find two numbers: $-40$ and $-8$: $-40×(-8)=320$, $-40 + (-8) = -48$

Split:
$$
5n^2 -40n -8n +64 = 5n(n - 8) -8(n - 8) = (5n - 8)(n - 8)
$$

✔
$$
-6(5n - 8)(n - 8)
$$

---

20) $ 10n^2 + 35n - 75 $

GCF: 5
$$
5(2n^2 + 7n - 15)
$$

AC = $2×(-15) = -30$, numbers: $10$ and $-3$: $10 + (-3) = 7$

Split:
$$
2n^2 +10n -3n -15 = 2n(n + 5) -3(n + 5) = (2n - 3)(n + 5)
$$

✔
$$
5(2n - 3)(n + 5)
$$

---

21) $ 20x^2 + 195x + 280 $

GCF: 5
$$
5(4x^2 + 39x + 56)
$$

AC = $4×56 = 224$, numbers: $32$ and $7$: $32+7=39$

Split:
$$
4x^2 +32x +7x +56 = 4x(x + 8) +7(x + 8) = (4x + 7)(x + 8)
$$

✔
$$
5(4x + 7)(x + 8)
$$

---

22) $ 48m^2 + 150m - 168 $

GCF: 6
$$
6(8m^2 + 25m - 28)
$$

AC = $8×(-28) = -224$, numbers: $32$ and $-7$: $32 + (-7) = 25$

Split:
$$
8m^2 +32m -7m -28 = 8m(m + 4) -7(m + 4) = (8m - 7)(m + 4)
$$

✔
$$
6(8m - 7)(m + 4)
$$

---

23) $ 16m^2 + 38m + 12 $

GCF: 2
$$
2(8m^2 + 19m + 6)
$$

AC = $8×6 = 48$, numbers: $16$ and $3$: $16+3=19$

Split:
$$
8m^2 +16m +3m +6 = 8m(m + 2) +3(m + 2) = (8m + 3)(m + 2)
$$

✔
$$
2(8m + 3)(m + 2)
$$

---

24) $ -12n^2 + 45n + 162 $

GCF: -3
$$
-3(4n^2 - 15n - 54)
$$

AC = $4×(-54) = -216$, numbers: $-27$ and $8$: $-27 + 8 = -19$ → no

Try: $-24$ and $9$: $-24×9 = -216$, $-24+9 = -15$ ✔

Split:
$$
4n^2 -24n +9n -54 = 4n(n - 6) +9(n - 6) = (4n + 9)(n - 6)
$$

✔
$$
-3(4n + 9)(n - 6)
$$

---

25) $ 12k^2 - 111k + 27 $

GCF: 3
$$
3(4k^2 - 37k + 9)
$$

AC = $4×9 = 36$, numbers: $-36$ and $-1$: $-36 + (-1) = -37$

Split:
$$
4k^2 -36k -k +9 = 4k(k - 9) -1(k - 9) = (4k - 1)(k - 9)
$$

✔
$$
3(4k - 1)(k - 9)
$$

---

26) $ -18k^2 + 15k + 168 $

GCF: -3
$$
-3(6k^2 - 5k - 56)
$$

AC = $6×(-56) = -336$, numbers: $-21$ and $16$: $-21×16 = -336$, $-21+16 = -5$

Split:
$$
6k^2 -21k +16k -56 = 3k(2k - 7) +8(2k - 7) = (3k + 8)(2k - 7)
$$

✔
$$
-3(3k + 8)(2k - 7)
$$

---

27) $ 7a^3 + 2a^2 + 42a + 12 $

Group terms:
$$
(7a^3 + 2a^2) + (42a + 12) = a^2(7a + 2) + 6(7a + 2)
$$

Factor:
$$
(7a + 2)(a^2 + 6)
$$

✔
$$
(7a + 2)(a^2 + 6)
$$

---

28) $ 5p^3 + 10p^2 - 7p - 14 $

Group:
$$
(5p^3 + 10p^2) + (-7p - 14) = 5p^2(p + 2) -7(p + 2) = (5p^2 - 7)(p + 2)
$$

✔
$$
(5p^2 - 7)(p + 2)
$$

---

29) $ 4m^3 + 8m^2 + 7m + 14 $

Group:
$$
(4m^3 + 8m^2) + (7m + 14) = 4m^2(m + 2) + 7(m + 2) = (4m^2 + 7)(m + 2)
$$

✔
$$
(4m^2 + 7)(m + 2)
$$

---

30) $ 16a^3 - 6a^2 - 8a + 3 $

Group:
$$
(16a^3 - 6a^2) + (-8a + 3) = 2a^2(8a - 3) -1(8a - 3) = (2a^2 - 1)(8a - 3)
$$

✔
$$
(2a^2 - 1)(8a - 3)
$$

---

31) $ 180p^3 - 30p^2 + 216p - 36 $

GCF: 6
$$
6(30p^3 - 5p^2 + 36p - 6)
$$

Group inside:
$$
(30p^3 - 5p^2) + (36p - 6) = 5p^2(6p - 1) + 6(6p - 1) = (5p^2 + 6)(6p - 1)
$$

✔
$$
6(5p^2 + 6)(6p - 1)
$$

---

32) $ 392m^3 - 112m^2 - 49m + 14 $

GCF: 7
$$
7(56m^3 - 16m^2 - 7m + 2)
$$

Group:
$$
(56m^3 - 16m^2) + (-7m + 2) = 8m^2(7m - 2) -1(7m - 2) = (8m^2 - 1)(7m - 2)
$$

Note: $8m^2 - 1 = (2\sqrt{2}m)^2 - 1^2$, but not rational factors.

Wait — actually, $8m^2 - 1$ is difference of squares? No: $8m^2 - 1 = (\sqrt{8}m)^2 - 1^2$, but irrational.

But we can leave it unless factored further.

Actually: $8m^2 - 1$ doesn't factor over integers.

So final answer:
$$
7(8m^2 - 1)(7m - 2)
$$

But $8m^2 - 1$ is not factorable with integer coefficients.

Wait: Is there a mistake?

Check: Try factoring $56m^3 - 16m^2 - 7m + 2$

Try grouping differently?

Alternatively, check if it factors as $(am^2 + bm + c)(dm + e)$

Try rational roots: possible roots: ±1,±2,±1/2,±1/4,etc.

Try $m = 1/2$:
$56(1/8) - 16(1/4) -7(1/2) +2 = 7 - 4 - 3.5 + 2 = 1.5 ≠ 0$

Try $m = 1/4$: too messy.

Alternatively, maybe we made an error in grouping.

Wait: original grouping was correct.

But let's double-check:

$$
(56m^3 - 16m^2) = 8m^2(7m - 2), \quad (-7m + 2) = -1(7m - 2)
$$

Yes! So:
$$
(8m^2 - 1)(7m - 2)
$$

But $8m^2 - 1$ is not factorable over integers. So we stop here.

✔
$$
7(8m^2 - 1)(7m - 2)
$$

---

33) $ 343p^3 + 196p^2 + 196p + 112 $

GCF: 7
$$
7(49p^3 + 28p^2 + 28p + 16)
$$

Group:
$$
(49p^3 + 28p^2) + (28p + 16) = 7p^2(7p + 4) + 4(7p + 4) = (7p^2 + 4)(7p + 4)
$$

✔
$$
7(7p^2 + 4)(7p + 4)
$$

---

34) $ 8m^3 + 32m^2 - 8m - 32 $

GCF: 8
$$
8(m^3 + 4m^2 - m - 4)
$$

Group:
$$
(m^3 + 4m^2) + (-m - 4) = m^2(m + 4) -1(m + 4) = (m^2 - 1)(m + 4)
$$

Now $m^2 - 1 = (m - 1)(m + 1)$

✔
$$
8(m - 1)(m + 1)(m + 4)
$$

---

## ✔ Final Answers (All Factored Completely):

1. $(b - 9)(b - 10)$
2. $(b + 6)(b - 4)$
3. $(x + 10)(x + 3)$
4. $(p - 10)(p + 7)$
5. $4(k + 8)(k - 3)$
6. $4(n - 9)(n + 7)$
7. $6(n + 8)(n - 1)$
8. $4(x + 5)(x - 2)$
9. $3(2x - 1)(x - 7)$
10. $3(5b - 6)(b + 5)$
11. $6(7a + 4)(a - 9)$
12. $3(7a - 10)(a + 3)$
13. $3(5r + 2)(r + 1)$
14. $2(5a + 8)(a - 7)$
15. $3(3r + 2)(r + 5)$
16. $3(k - 1)(3k + 4)$
17. $5(3n - 10)(n - 4)$
18. $3(2p - 1)(p + 6)$
19. $-6(5n - 8)(n - 8)$
20. $5(2n - 3)(n + 5)$
21. $5(4x + 7)(x + 8)$
22. $6(8m - 7)(m + 4)$
23. $2(8m + 3)(m + 2)$
24. $-3(4n + 9)(n - 6)$
25. $3(4k - 1)(k - 9)$
26. $-3(3k + 8)(2k - 7)$
27. $(7a + 2)(a^2 + 6)$
28. $(5p^2 - 7)(p + 2)$
29. $(4m^2 + 7)(m + 2)$
30. $(2a^2 - 1)(8a - 3)$
31. $6(5p^2 + 6)(6p - 1)$
32. $7(8m^2 - 1)(7m - 2)$
33. $7(7p^2 + 4)(7p + 4)$
34. $8(m - 1)(m + 1)(m + 4)$

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