Printed Algebra 2 worksheet covering rational expressions rests on a desk next to a computer keyboard.

Let's solve each problem step-by-step, starting with multiplying rational expressions, then dividing, and finally mixed operations. For each, we will:

1. Simplify the expression.
2. State the excluded values (values that make any denominator zero).

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Multiplying Rational Expressions

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#### 1)
$$
\frac{1}{x - 4} \cdot \frac{-x^2 + 9x - 20}{x + 2}
$$

Step 1: Factor numerator:
$-x^2 + 9x - 20 = -(x^2 - 9x + 20) = -(x - 5)(x - 4)$

So:
$$
\frac{1}{x - 4} \cdot \frac{-(x - 5)(x - 4)}{x + 2}
$$

Cancel $x - 4$:
$$
= \frac{-(x - 5)}{x + 2}
$$

Answer: $\boxed{\frac{-(x - 5)}{x + 2}}$

Excluded values: $x \neq 4, -2$ (denominators zero)

---

#### 2)
$$
\frac{7}{b^2 + 5b + 4} \cdot \frac{b^2 + 8b + 7}{b + 7}
$$

Factor:
- $b^2 + 5b + 4 = (b + 1)(b + 4)$
- $b^2 + 8b + 7 = (b + 1)(b + 7)$

So:
$$
\frac{7}{(b+1)(b+4)} \cdot \frac{(b+1)(b+7)}{b+7}
$$

Cancel $(b+1)$ and $(b+7)$:
$$
= \frac{7}{b+4}
$$

Answer: $\boxed{\frac{7}{b+4}}$

Excluded values: $b \neq -1, -4, -7$

---

#### 3)
$$
\frac{9r^2 + 18r}{r + 2} \cdot \frac{1}{r - 2}
$$

Factor numerator: $9r(r + 2)$

So:
$$
\frac{9r(r + 2)}{r + 2} \cdot \frac{1}{r - 2} = 9r \cdot \frac{1}{r - 2} = \frac{9r}{r - 2}
$$

Answer: $\boxed{\frac{9r}{r - 2}}$

Excluded values: $r \neq -2, 2$

---

#### 4)
$$
\frac{8n}{4n + 4} \cdot \frac{4n + 4}{10}
$$

Factor: $4n + 4 = 4(n + 1)$

So:
$$
\frac{8n}{4(n+1)} \cdot \frac{4(n+1)}{10} = \frac{8n}{4(n+1)} \cdot \frac{4(n+1)}{10}
$$

Cancel $4(n+1)$:
$$
= \frac{8n}{10} = \frac{4n}{5}
$$

Answer: $\boxed{\frac{4n}{5}}$

Excluded values: $n \neq -1$ (since $4n + 4 = 0$ when $n = -1$)

---

#### 5)
$$
\frac{7a^2 - 28a}{a - 4} \cdot \frac{1}{5a^2}
$$

Factor: $7a(a - 4)$

So:
$$
\frac{7a(a - 4)}{a - 4} \cdot \frac{1}{5a^2} = 7a \cdot \frac{1}{5a^2} = \frac{7}{5a}
$$

Answer: $\boxed{\frac{7}{5a}}$

Excluded values: $a \neq 4, 0$

---

#### 6)
$$
\frac{1}{x - 5} \cdot \frac{6x - 30}{6}
$$

Factor: $6x - 30 = 6(x - 5)$

So:
$$
\frac{1}{x - 5} \cdot \frac{6(x - 5)}{6} = \frac{1}{x - 5} \cdot (x - 5) = 1
$$

Answer: $\boxed{1}$

Excluded values: $x \neq 5$

---

#### 7)
$$
\frac{1}{v - 4} \cdot \frac{v^2 - 7v + 12}{v + 3}
$$

Factor: $v^2 - 7v + 12 = (v - 3)(v - 4)$

So:
$$
\frac{1}{v - 4} \cdot \frac{(v - 3)(v - 4)}{v + 3} = \frac{v - 3}{v + 3}
$$

Answer: $\boxed{\frac{v - 3}{v + 3}}$

Excluded values: $v \neq 4, -3$

---

#### 8)
$$
\frac{n - 2}{n^2 - 13n + 40} \cdot (n - 5)
$$

Factor denominator: $n^2 - 13n + 40 = (n - 5)(n - 8)$

So:
$$
\frac{n - 2}{(n - 5)(n - 8)} \cdot (n - 5) = \frac{n - 2}{n - 8}
$$

Answer: $\boxed{\frac{n - 2}{n - 8}}$

Excluded values: $n \neq 5, 8$

---

#### 9)
$$
\frac{k - 4}{27k^2 + 27k} \cdot (3k + 3)
$$

Factor:
- $27k^2 + 27k = 27k(k + 1)$
- $3k + 3 = 3(k + 1)$

So:
$$
\frac{k - 4}{27k(k + 1)} \cdot 3(k + 1) = \frac{k - 4}{27k} \cdot 3 = \frac{3(k - 4)}{27k} = \frac{k - 4}{9k}
$$

Answer: $\boxed{\frac{k - 4}{9k}}$

Excluded values: $k \neq 0, -1$

---

#### 10)
$$
\frac{50x^3 + 10x^2}{50x + 10} \cdot \frac{5}{10x^2}
$$

Factor:
- Numerator: $10x^2(5x + 1)$
- Denominator: $10(5x + 1)$

So:
$$
\frac{10x^2(5x + 1)}{10(5x + 1)} \cdot \frac{5}{10x^2} = x^2 \cdot \frac{5}{10x^2} = \frac{5}{10} = \frac{1}{2}
$$

Answer: $\boxed{\frac{1}{2}}$

Excluded values: $x \neq -\frac{1}{5}, 0$

---

#### 11)
$$
\frac{1}{x + 8} \cdot \frac{18x^2 - 12x}{3x - 2}
$$

Factor numerator: $6x(3x - 2)$

So:
$$
\frac{1}{x + 8} \cdot \frac{6x(3x - 2)}{3x - 2} = \frac{1}{x + 8} \cdot 6x = \frac{6x}{x + 8}
$$

Answer: $\boxed{\frac{6x}{x + 8}}$

Excluded values: $x \neq -8, \frac{2}{3}$

---

#### 12)
$$
\frac{5n + 2}{45n^2 + 18n} \cdot (n - 1)
$$

Factor denominator: $9n(5n + 2)$

So:
$$
\frac{5n + 2}{9n(5n + 2)} \cdot (n - 1) = \frac{1}{9n} \cdot (n - 1) = \frac{n - 1}{9n}
$$

Answer: $\boxed{\frac{n - 1}{9n}}$

Excluded values: $n \neq 0, -\frac{2}{5}$

---

Dividing Rational Expressions

> Recall: Division → Multiply by reciprocal.

---

#### 13)
$$
\frac{m - 4}{3m^2 + 24m} \div \frac{m - 4}{m - 6}
$$

First, factor: $3m^2 + 24m = 3m(m + 8)$

Now:
$$
\frac{m - 4}{3m(m + 8)} \div \frac{m - 4}{m - 6} = \frac{m - 4}{3m(m + 8)} \cdot \frac{m - 6}{m - 4}
$$

Cancel $m - 4$:
$$
= \frac{1}{3m(m + 8)} \cdot (m - 6) = \frac{m - 6}{3m(m + 8)}
$$

Answer: $\boxed{\frac{m - 6}{3m(m + 8)}}$

Excluded values: $m \neq 0, -8, 4, 6$

---

#### 14)
$$
\frac{n^2 - 9n - 10}{n + 1} \div \frac{n + 3}{n + 10}
$$

Factor: $n^2 - 9n - 10 = (n - 10)(n + 1)$

So:
$$
\frac{(n - 10)(n + 1)}{n + 1} \div \frac{n + 3}{n + 10} = (n - 10) \div \frac{n + 3}{n + 10} = (n - 10) \cdot \frac{n + 10}{n + 3}
$$

$$
= \frac{(n - 10)(n + 10)}{n + 3} = \frac{n^2 - 100}{n + 3}
$$

Answer: $\boxed{\frac{n^2 - 100}{n + 3}}$

Excluded values: $n \neq -1, -10, -3$

---

#### 15)
$$
\frac{p - 7}{8} \div \frac{p + 5}{8p + 8}
$$

Factor: $8p + 8 = 8(p + 1)$

So:
$$
\frac{p - 7}{8} \div \frac{p + 5}{8(p + 1)} = \frac{p - 7}{8} \cdot \frac{8(p + 1)}{p + 5}
$$

Cancel 8:
$$
= (p - 7) \cdot \frac{p + 1}{p + 5} = \frac{(p - 7)(p + 1)}{p + 5}
$$

Answer: $\boxed{\frac{(p - 7)(p + 1)}{p + 5}}$

Excluded values: $p \neq -5, -1$

---

#### 16)
$$
\frac{4}{20x - 12} \div \frac{1}{20x^2 - 12x}
$$

Factor:
- $20x - 12 = 4(5x - 3)$
- $20x^2 - 12x = 4x(5x - 3)$

So:
$$
\frac{4}{4(5x - 3)} \div \frac{1}{4x(5x - 3)} = \frac{1}{5x - 3} \div \frac{1}{4x(5x - 3)} = \frac{1}{5x - 3} \cdot 4x(5x - 3)
$$

Cancel $5x - 3$:
$$
= 4x
$$

Answer: $\boxed{4x}$

Excluded values: $x \neq 0, \frac{3}{5}$

---

#### 17)
$$
(n - 6) \div \frac{n^2 - 9n + 18}{3}
$$

Factor: $n^2 - 9n + 18 = (n - 3)(n - 6)$

So:
$$
(n - 6) \div \frac{(n - 3)(n - 6)}{3} = (n - 6) \cdot \frac{3}{(n - 3)(n - 6)} = \frac{3}{n - 3}
$$

Answer: $\boxed{\frac{3}{n - 3}}$

Excluded values: $n \neq 3, 6$

---

#### 18)
$$
\frac{1}{7b + 35} \div \frac{3b}{b^2 + 11b + 30}
$$

Factor:
- $7b + 35 = 7(b + 5)$
- $b^2 + 11b + 30 = (b + 5)(b + 6)$

So:
$$
\frac{1}{7(b + 5)} \div \frac{3b}{(b + 5)(b + 6)} = \frac{1}{7(b + 5)} \cdot \frac{(b + 5)(b + 6)}{3b}
$$

Cancel $b + 5$:
$$
= \frac{1}{7} \cdot \frac{b + 6}{3b} = \frac{b + 6}{21b}
$$

Answer: $\boxed{\frac{b + 6}{21b}}$

Excluded values: $b \neq -5, -6, 0$

---

#### 19)
$$
\frac{9r + 90}{4} \div \frac{9r + 90}{4r^2}
$$

Factor: $9r + 90 = 9(r + 10)$

So:
$$
\frac{9(r + 10)}{4} \div \frac{9(r + 10)}{4r^2} = \frac{9(r + 10)}{4} \cdot \frac{4r^2}{9(r + 10)} = r^2
$$

Answer: $\boxed{r^2}$

Excluded values: $r \neq -10$

---

#### 20)
$$
\frac{x + 7}{x + 6} \div \frac{6x - 60}{x - 10}
$$

Factor: $6x - 60 = 6(x - 10)$

So:
$$
\frac{x + 7}{x + 6} \div \frac{6(x - 10)}{x - 10} = \frac{x + 7}{x + 6} \div 6 = \frac{x + 7}{x + 6} \cdot \frac{1}{6} = \frac{x + 7}{6(x + 6)}
$$

Answer: $\boxed{\frac{x + 7}{6(x + 6)}}$

Excluded values: $x \neq -6, 10$

---

#### 21)
$$
\frac{1}{8 - 5x} \div \frac{5x^2}{15x^2 - 24x}
$$

Note: $8 - 5x = -(5x - 8)$

Factor denominator: $15x^2 - 24x = 3x(5x - 8)$

So:
$$
\frac{1}{8 - 5x} \div \frac{5x^2}{3x(5x - 8)} = \frac{1}{-(5x - 8)} \div \frac{5x^2}{3x(5x - 8)}
$$

$$
= \frac{-1}{5x - 8} \cdot \frac{3x(5x - 8)}{5x^2} = -1 \cdot \frac{3x}{5x^2} = -\frac{3}{5x}
$$

Answer: $\boxed{-\frac{3}{5x}}$

Excluded values: $x \neq 0, \frac{8}{5}$

---

#### 22)
$$
\frac{6a^3 + 2a^2}{a + 3} \div (3a + 1)
$$

Factor numerator: $2a^2(3a + 1)$

So:
$$
\frac{2a^2(3a + 1)}{a + 3} \div (3a + 1) = \frac{2a^2(3a + 1)}{a + 3} \cdot \frac{1}{3a + 1} = \frac{2a^2}{a + 3}
$$

Answer: $\boxed{\frac{2a^2}{a + 3}}$

Excluded values: $a \neq -3, -\frac{1}{3}$

---

#### 23)
$$
\frac{7v + 5}{7} \div \frac{14v^2 + 45v + 25}{4v^3 + 10v^2}
$$

Factor:
- $14v^2 + 45v + 25$: Try factoring: $(2v + 5)(7v + 5)$ → Check: $2v*7v = 14v^2$, $2v*5 = 10v$, $5*7v = 35v$, total $45v$, yes!
- $4v^3 + 10v^2 = 2v^2(2v + 5)$

So:
$$
\frac{7v + 5}{7} \div \frac{(2v + 5)(7v + 5)}{2v^2(2v + 5)} = \frac{7v + 5}{7} \div \frac{7v + 5}{2v^2}
$$

$$
= \frac{7v + 5}{7} \cdot \frac{2v^2}{7v + 5} = \frac{1}{7} \cdot 2v^2 = \frac{2v^2}{7}
$$

Answer: $\boxed{\frac{2v^2}{7}}$

Excluded values: $v \neq 0, -\frac{5}{2}$

---

#### 24)
$$
\frac{3n^2 - 27n - 30}{10n} \div \frac{24n + 24}{10n}
$$

Factor:
- $3n^2 - 27n - 30 = 3(n^2 - 9n - 10) = 3(n - 10)(n + 1)$
- $24n + 24 = 24(n + 1)$

So:
$$
\frac{3(n - 10)(n + 1)}{10n} \div \frac{24(n + 1)}{10n} = \frac{3(n - 10)(n + 1)}{10n} \cdot \frac{10n}{24(n + 1)}
$$

Cancel $10n$, $(n + 1)$:
$$
= \frac{3(n - 10)}{24} = \frac{n - 10}{8}
$$

Answer: $\boxed{\frac{n - 10}{8}}$

Excluded values: $n \neq 0, -1$

---

Mixed Operations (Multiply or Divide)

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#### 25)
$$
\frac{3x^2 + 6x}{6x^2 - 3x} \div \frac{x + 1}{2x - 1}
$$

Factor:
- $3x^2 + 6x = 3x(x + 2)$
- $6x^2 - 3x = 3x(2x - 1)$

So:
$$
\frac{3x(x + 2)}{3x(2x - 1)} \div \frac{x + 1}{2x - 1} = \frac{x + 2}{2x - 1} \div \frac{x + 1}{2x - 1}
$$

$$
= \frac{x + 2}{2x - 1} \cdot \frac{2x - 1}{x + 1} = \frac{x + 2}{x + 1}
$$

Answer: $\boxed{\frac{x + 2}{x + 1}}$

Excluded values: $x \neq 0, \frac{1}{2}, -1$

---

#### 26)
$$
\frac{8k^3 - 16k^2}{k - 5} \div \frac{40k^3 + 16k^2}{5k + 2}
$$

Factor:
- $8k^3 - 16k^2 = 8k^2(k - 2)$
- $40k^3 + 16k^2 = 8k^2(5k + 2)$

So:
$$
\frac{8k^2(k - 2)}{k - 5} \div \frac{8k^2(5k + 2)}{5k + 2} = \frac{8k^2(k - 2)}{k - 5} \div 8k^2 = \frac{8k^2(k - 2)}{k - 5} \cdot \frac{1}{8k^2} = \frac{k - 2}{k - 5}
$$

Answer: $\boxed{\frac{k - 2}{k - 5}}$

Excluded values: $k \neq 5, 0, -\frac{2}{5}$

---

#### 27)
$$
\frac{15p + 30}{p + 2} \div \frac{15p + 30}{p + 5}
$$

Factor: $15p + 30 = 15(p + 2)$

So:
$$
\frac{15(p + 2)}{p + 2} \div \frac{15(p + 2)}{p + 5} = 15 \div \frac{15(p + 2)}{p + 5} = 15 \cdot \frac{p + 5}{15(p + 2)} = \frac{p + 5}{p + 2}
$$

Answer: $\boxed{\frac{p + 5}{p + 2}}$

Excluded values: $p \neq -2, -5$

---

#### 28)
$$
\frac{50n}{2n^2 + 18n + 28} \cdot (2n + 4)
$$

Factor:
- $2n^2 + 18n + 28 = 2(n^2 + 9n + 14) = 2(n + 2)(n + 7)$
- $2n + 4 = 2(n + 2)$

So:
$$
\frac{50n}{2(n + 2)(n + 7)} \cdot 2(n + 2) = \frac{50n}{2(n + 2)(n + 7)} \cdot 2(n + 2)
$$

Cancel $2$ and $(n + 2)$:
$$
= \frac{50n}{n + 7}
$$

Answer: $\boxed{\frac{50n}{n + 7}}$

Excluded values: $n \neq -2, -7$

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✔ Final Summary of Answers

| Problem | Answer |
|--------|--------|
| 1 | $\frac{-(x - 5)}{x + 2}$ |
| 2 | $\frac{7}{b + 4}$ |
| 3 | $\frac{9r}{r - 2}$ |
| 4 | $\frac{4n}{5}$ |
| 5 | $\frac{7}{5a}$ |
| 6 | $1$ |
| 7 | $\frac{v - 3}{v + 3}$ |
| 8 | $\frac{n - 2}{n - 8}$ |
| 9 | $\frac{k - 4}{9k}$ |
| 10 | $\frac{1}{2}$ |
| 11 | $\frac{6x}{x + 8}$ |
| 12 | $\frac{n - 1}{9n}$ |
| 13 | $\frac{m - 6}{3m(m + 8)}$ |
| 14 | $\frac{n^2 - 100}{n + 3}$ |
| 15 | $\frac{(p - 7)(p + 1)}{p + 5}$ |
| 16 | $4x$ |
| 17 | $\frac{3}{n - 3}$ |
| 18 | $\frac{b + 6}{21b}$ |
| 19 | $r^2$ |
| 20 | $\frac{x + 7}{6(x + 6)}$ |
| 21 | $-\frac{3}{5x}$ |
| 22 | $\frac{2a^2}{a + 3}$ |
| 23 | $\frac{2v^2}{7}$ |
| 24 | $\frac{n - 10}{8}$ |
| 25 | $\frac{x + 2}{x + 1}$ |
| 26 | $\frac{k - 2}{k - 5}$ |
| 27 | $\frac{p + 5}{p + 2}$ |
| 28 | $\frac{50n}{n + 7}$ |

All excluded values are listed in each solution. Let me know if you want them compiled in a table!