Math worksheet for dividing rational expressions, including algebraic fractions and polynomial division problems.

To solve the given problems, we need to divide rational expressions. The general steps for dividing rational expressions are:

1. Rewrite the division as multiplication by the reciprocal.
2. Factor all polynomials in the numerators and denominators.
3. Simplify by canceling out common factors.
4. Write the final simplified expression.

Let's solve each problem step by step.

---

Problem 1:

\[
\frac{x^2 - 16}{x^2 - 10x + 25} \div \frac{3x - 12}{x^2 - 3x - 10}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{x^2 - 16}{x^2 - 10x + 25} \cdot \frac{x^2 - 3x - 10}{3x - 12}
\]

#### Step 2: Factor all polynomials.
- \(x^2 - 16 = (x - 4)(x + 4)\)
- \(x^2 - 10x + 25 = (x - 5)^2\)
- \(x^2 - 3x - 10 = (x - 5)(x + 2)\)
- \(3x - 12 = 3(x - 4)\)

So the expression becomes:
\[
\frac{(x - 4)(x + 4)}{(x - 5)^2} \cdot \frac{(x - 5)(x + 2)}{3(x - 4)}
\]

#### Step 3: Simplify by canceling out common factors.
- Cancel \((x - 4)\) from the numerator and denominator.
- Cancel one \((x - 5)\) from the numerator and denominator.

\[
\frac{(x + 4)}{(x - 5)} \cdot \frac{(x + 2)}{3} = \frac{(x + 4)(x + 2)}{3(x - 5)}
\]

#### Final Answer:
\[
\boxed{\frac{(x + 4)(x + 2)}{3(x - 5)}}
\]

---

Problem 2:

\[
\frac{x^3 + 9x + 14}{x^2 - 49} \div \frac{3x + 6}{x^2 + x - 56}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{x^3 + 9x + 14}{x^2 - 49} \cdot \frac{x^2 + x - 56}{3x + 6}
\]

#### Step 2: Factor all polynomials.
- \(x^2 - 49 = (x - 7)(x + 7)\)
- \(3x + 6 = 3(x + 2)\)
- \(x^2 + x - 56 = (x + 8)(x - 7)\)

The cubic polynomial \(x^3 + 9x + 14\) does not factor nicely over the integers, so we leave it as is for now.

So the expression becomes:
\[
\frac{x^3 + 9x + 14}{(x - 7)(x + 7)} \cdot \frac{(x + 8)(x - 7)}{3(x + 2)}
\]

#### Step 3: Simplify by canceling out common factors.
- Cancel \((x - 7)\) from the numerator and denominator.

\[
\frac{x^3 + 9x + 14}{(x + 7)} \cdot \frac{(x + 8)}{3(x + 2)} = \frac{(x^3 + 9x + 14)(x + 8)}{3(x + 7)(x + 2)}
\]

#### Final Answer:
\[
\boxed{\frac{(x^3 + 9x + 14)(x + 8)}{3(x + 7)(x + 2)}}
\]

---

Problem 3:

\[
\frac{x^2 - x - 2}{2x^2 - 5x + 2} \div \frac{x^3 - x - 12}{2x^2 + 5x - 3}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{x^2 - x - 2}{2x^2 - 5x + 2} \cdot \frac{2x^2 + 5x - 3}{x^3 - x - 12}
\]

#### Step 2: Factor all polynomials.
- \(x^2 - x - 2 = (x - 2)(x + 1)\)
- \(2x^2 - 5x + 2 = (2x - 1)(x - 2)\)
- \(2x^2 + 5x - 3 = (2x - 1)(x + 3)\)
- \(x^3 - x - 12\) does not factor nicely over the integers, so we leave it as is.

So the expression becomes:
\[
\frac{(x - 2)(x + 1)}{(2x - 1)(x - 2)} \cdot \frac{(2x - 1)(x + 3)}{x^3 - x - 12}
\]

#### Step 3: Simplify by canceling out common factors.
- Cancel \((x - 2)\) from the numerator and denominator.
- Cancel \((2x - 1)\) from the numerator and denominator.

\[
\frac{(x + 1)}{1} \cdot \frac{(x + 3)}{x^3 - x - 12} = \frac{(x + 1)(x + 3)}{x^3 - x - 12}
\]

#### Final Answer:
\[
\boxed{\frac{(x + 1)(x + 3)}{x^3 - x - 12}}
\]

---

Problem 4:

\[
\frac{2}{3x} \div \frac{7}{8x}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{2}{3x} \cdot \frac{8x}{7}
\]

#### Step 2: Simplify.
- Cancel \(x\) from the numerator and denominator.

\[
\frac{2 \cdot 8}{3 \cdot 7} = \frac{16}{21}
\]

#### Final Answer:
\[
\boxed{\frac{16}{21}}
\]

---

Problem 5:

\[
\frac{x - 4}{x^2 - 2x - 8} \div \frac{1}{x - 5}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{x - 4}{x^2 - 2x - 8} \cdot (x - 5)
\]

#### Step 2: Factor all polynomials.
- \(x^2 - 2x - 8 = (x - 4)(x + 2)\)

So the expression becomes:
\[
\frac{x - 4}{(x - 4)(x + 2)} \cdot (x - 5)
\]

#### Step 3: Simplify by canceling out common factors.
- Cancel \((x - 4)\) from the numerator and denominator.

\[
\frac{1}{x + 2} \cdot (x - 5) = \frac{x - 5}{x + 2}
\]

#### Final Answer:
\[
\boxed{\frac{x - 5}{x + 2}}
\]

---

Problem 6:

\[
\frac{20x}{16} \div \frac{2x^3}{3x}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{20x}{16} \cdot \frac{3x}{2x^3}
\]

#### Step 2: Simplify.
- Simplify \(\frac{20x}{16}\) to \(\frac{5x}{4}\).
- Cancel \(x\) from the numerator and denominator.

\[
\frac{5x}{4} \cdot \frac{3}{2x^2} = \frac{5 \cdot 3}{4 \cdot 2x} = \frac{15}{8x}
\]

#### Final Answer:
\[
\boxed{\frac{15}{8x}}
\]

---

Problem 7:

\[
\frac{7x^2}{7x^3 + 56x^2} \div \frac{2}{x^2 + 7x - 8}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{7x^2}{7x^3 + 56x^2} \cdot \frac{x^2 + 7x - 8}{2}
\]

#### Step 2: Factor all polynomials.
- \(7x^3 + 56x^2 = 7x^2(x + 8)\)
- \(x^2 + 7x - 8 = (x + 8)(x - 1)\)

So the expression becomes:
\[
\frac{7x^2}{7x^2(x + 8)} \cdot \frac{(x + 8)(x - 1)}{2}
\]

#### Step 3: Simplify by canceling out common factors.
- Cancel \(7x^2\) from the numerator and denominator.
- Cancel \((x + 8)\) from the numerator and denominator.

\[
\frac{1}{1} \cdot \frac{(x - 1)}{2} = \frac{x - 1}{2}
\]

#### Final Answer:
\[
\boxed{\frac{x - 1}{2}}
\]

---

Problem 8:

\[
\frac{2x^2 - x - 15}{x^2 - 2x - 3} \div \frac{2x^2 + 3x - 5}{1 - x^2}
\]

#### Step 1: Rewrite the division as multiplication by the reciprocal.
\[
\frac{2x^2 - x - 15}{x^2 - 2x - 3} \cdot \frac{1 - x^2}{2x^2 + 3x - 5}
\]

#### Step 2: Factor all polynomials.
- \(2x^2 - x - 15 = (2x + 5)(x - 3)\)
- \(x^2 - 2x - 3 = (x - 3)(x + 1)\)
- \(2x^2 + 3x - 5 = (2x - 5)(x + 1)\)
- \(1 - x^2 = (1 - x)(1 + x) = -(x - 1)(x + 1)\)

So the expression becomes:
\[
\frac{(2x + 5)(x - 3)}{(x - 3)(x + 1)} \cdot \frac{-(x - 1)(x + 1)}{(2x - 5)(x + 1)}
\]

#### Step 3: Simplify by canceling out common factors.
- Cancel \((x - 3)\) from the numerator and denominator.
- Cancel \((x + 1)\) from the numerator and denominator.

\[
\frac{(2x + 5)}{1} \cdot \frac{-(x - 1)}{(2x - 5)} = \frac{-(2x + 5)(x - 1)}{2x - 5}
\]

#### Final Answer:
\[
\boxed{\frac{-(2x + 5)(x - 1)}{2x - 5}}
\]

---

Final Answers:

1. \(\boxed{\frac{(x + 4)(x + 2)}{3(x - 5)}}\)
2. \(\boxed{\frac{(x^3 + 9x + 14)(x + 8)}{3(x + 7)(x + 2)}}\)
3. \(\boxed{\frac{(x + 1)(x + 3)}{x^3 - x - 12}}\)
4. \(\boxed{\frac{16}{21}}\)
5. \(\boxed{\frac{x - 5}{x + 2}}\)
6. \(\boxed{\frac{15}{8x}}\)
7. \(\boxed{\frac{x - 1}{2}}\)
8. \(\boxed{\frac{-(2x + 5)(x - 1)}{2x - 5}}\)