31) $$ 7 n^{3}+21 n^{2}-5 n-15 $$ 33) $$ 15 a b-2 b^{2}-6 a+5 b^{3} $$ 35) $$ 5 m n-10+2 m-25 n $$ 37) $$ 4 u v+14 u^{2}+12 v+42 u $$ 3) $$ 56 a b+14-49 a-16 b $$

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UpStudy AI · Accepted Answer

**31) Factor $$ 7 n^{3}+21 n^{2}-5 n-15 $$**

1. Group the terms:
   $$
   (7n^3 + 21n^2) + (-5n - 15)
   $$
2. Factor each group:
   $$
   7n^2(n+3) - 5(n+3)
   $$
3. Factor out the common factor $$ (n+3) $$:
   $$
   (n+3)(7n^2 - 5)
   $$

---

**33) Factor $$ 15ab-2b^{2}-6a+5b^{3} $$**

1. Rearrange the terms:
   $$
   15ab -6a -2b^2 +5b^3
   $$
2. Group the terms:
   $$
   (15ab - 6a) + (-2b^2 + 5b^3)
   $$
3. Factor each group:
   - From $$ 15ab - 6a $$, factor $$ 3a $$:
     $$
     3a(5b-2)
     $$
   - From $$ -2b^2 + 5b^3 $$, factor $$ b^2 $$:
     $$
     b^2(5b-2)
     $$
4. Factor out the common binomial $$ (5b-2) $$:
   $$
   (5b-2)(3a + b^2)
   $$

---

**35) Factor $$ 5mn-10+2m-25n $$**

1. Rearrange the terms to group like terms:
   $$
   5mn - 25n + 2m - 10
   $$
2. Group the terms:
   $$
   (5mn - 25n) + (2m - 10)
   $$
3. Factor each group:
   - From $$ 5mn - 25n $$, factor $$ 5n $$:
     $$
     5n(m-5)
     $$
   - From $$ 2m - 10 $$, factor $$ 2 $$:
     $$
     2(m-5)
     $$
4. Factor out the common binomial $$ (m-5) $$:
   $$
   (m-5)(5n + 2)
   $$

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**37) Factor $$ 4uv+14u^{2}+12v+42u $$**

1. Rearrange and group the terms:
   $$
   (4uv + 12v) + (14u^2 + 42u)
   $$
2. Factor each group:
   - From $$ 4uv + 12v $$, factor $$ 4v $$:
     $$
     4v(u+3)
     $$
   - From $$ 14u^2 + 42u $$, factor $$ 14u $$:
     $$
     14u(u+3)
     $$
3. Factor out the common binomial $$ (u+3) $$:
   $$
   (u+3)(4v + 14u)
   $$
4. Factor a common factor of $$ 2 $$ from the second binomial:
   $$
   (u+3) \cdot 2(2v + 7u) = 2(u+3)(2v+7u)
   $$

---

**3) Factor $$ 56ab+14-49a-16b $$**

1. Rearrange the terms:
   $$
   56ab - 49a - 16b + 14
   $$
2. Group the terms:
   $$
   (56ab - 49a) + (-16b + 14)
   $$
3. Factor each group:
   - From $$ 56ab - 49a $$, factor $$ 7a $$:
     $$
     7a(8b-7)
     $$
   - From $$ -16b + 14 $$, factor $$-2$$:
     $$
     -2(8b-7)
     $$
4. Factor out the common factor $$ (8b-7) $$:
   $$
   (8b-7)(7a-2)
   $$
**31) $$ 7n^{3}+21n^{2}-5n-15 = (n+3)(7n^{2}-5) $$** **33) $$ 15ab-2b^{2}-6a+5b^{3} = (5b-2)(3a+b^{2}) $$** **35) $$ 5mn-10+2m-25n = (m-5)(5n+2) $$** **37) $$ 4uv+14u^{2}+12v+42u = 2(u+3)(2v+7u) $$** **3) $$ 56ab+14-49a-16b = (8b-7)(7a-2) $$**

31) \( 7 n^{3}+21 n^{2}-5 n-15 \) 33) \( 15 a b-2 b^{2}-6 a+5 b^{3} \) 35) \( 5 m n-10+2 m-25 n \) 37) \( 4 u v+14 u^{2}+12 v+42 u \) 3) \( 56 a b+14-49 a-16 b \)

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