Simplify rational expressions worksheet with eight algebra problems involving multiplication and division of polynomials.

To solve the given problems involving multiplying and dividing rational expressions, we will simplify each expression step by step. Let's go through each problem:

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Problem 1:

Simplify:
\[
\frac{6x - 18}{4x} \cdot \frac{x}{2x - 6}
\]

#### Step 1: Factorize where possible
- \(6x - 18 = 6(x - 3)\)
- \(2x - 6 = 2(x - 3)\)

So the expression becomes:
\[
\frac{6(x - 3)}{4x} \cdot \frac{x}{2(x - 3)}
\]

#### Step 2: Simplify by canceling common factors
- The \(x - 3\) terms in the numerator and denominator cancel out.
- The \(x\) in the numerator and denominator also cancel out.

This leaves:
\[
\frac{6}{4} \cdot \frac{1}{2} = \frac{6}{8} = \frac{3}{4}
\]

#### Final Answer:
\[
\boxed{\frac{3}{4}}
\]

---

Problem 2:

Simplify:
\[
\frac{5x^2 - 15x}{2x^2 + 11x + 12} \cdot \frac{3x^2 - 48}{10x^3 - 70x^2 + 120x}
\]

#### Step 1: Factorize each term
- \(5x^2 - 15x = 5x(x - 3)\)
- \(2x^2 + 11x + 12 = (2x + 3)(x + 4)\)
- \(3x^2 - 48 = 3(x^2 - 16) = 3(x - 4)(x + 4)\)
- \(10x^3 - 70x^2 + 120x = 10x(x^2 - 7x + 12) = 10x(x - 3)(x - 4)\)

So the expression becomes:
\[
\frac{5x(x - 3)}{(2x + 3)(x + 4)} \cdot \frac{3(x - 4)(x + 4)}{10x(x - 3)(x - 4)}
\]

#### Step 2: Simplify by canceling common factors
- The \(x\) terms in the numerator and denominator cancel out.
- The \(x - 3\) terms in the numerator and denominator cancel out.
- The \(x + 4\) terms in the numerator and denominator cancel out.
- The \(x - 4\) terms in the numerator and denominator cancel out.

This leaves:
\[
\frac{5 \cdot 3}{(2x + 3) \cdot 10} = \frac{15}{10(2x + 3)} = \frac{3}{2(2x + 3)}
\]

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

---

Problem 3:

Simplify:
\[
\frac{7x + 14}{2x^2 - 8} \cdot (x^2 + 3x - 10)
\]

#### Step 1: Factorize each term
- \(7x + 14 = 7(x + 2)\)
- \(2x^2 - 8 = 2(x^2 - 4) = 2(x - 2)(x + 2)\)
- \(x^2 + 3x - 10 = (x + 5)(x - 2)\)

So the expression becomes:
\[
\frac{7(x + 2)}{2(x - 2)(x + 2)} \cdot (x + 5)(x - 2)
\]

#### Step 2: Simplify by canceling common factors
- The \(x + 2\) terms in the numerator and denominator cancel out.
- The \(x - 2\) terms in the numerator and denominator cancel out.

This leaves:
\[
\frac{7}{2} \cdot (x + 5) = \frac{7(x + 5)}{2}
\]

#### Final Answer:
\[
\boxed{\frac{7(x + 5)}{2}}
\]

---

Problem 4:

Simplify:
\[
\frac{x^3 - 27}{x^4 - 9x^2} \cdot \frac{x^5 + 3x^4}{x^2 + 3x + 9}
\]

#### Step 1: Factorize each term
- \(x^3 - 27 = (x - 3)(x^2 + 3x + 9)\) (difference of cubes)
- \(x^4 - 9x^2 = x^2(x^2 - 9) = x^2(x - 3)(x + 3)\)
- \(x^5 + 3x^4 = x^4(x + 3)\)

So the expression becomes:
\[
\frac{(x - 3)(x^2 + 3x + 9)}{x^2(x - 3)(x + 3)} \cdot \frac{x^4(x + 3)}{x^2 + 3x + 9}
\]

#### Step 2: Simplify by canceling common factors
- The \(x - 3\) terms in the numerator and denominator cancel out.
- The \(x^2 + 3x + 9\) terms in the numerator and denominator cancel out.
- The \(x + 3\) terms in the numerator and denominator cancel out.

This leaves:
\[
\frac{1}{x^2} \cdot x^4 = x^{4-2} = x^2
\]

#### Final Answer:
\[
\boxed{x^2}
\]

---

Problem 5:

Simplify:
\[
\frac{x^2 - 5x - 14}{28 + 3x - x^2}
\]

#### Step 1: Factorize each term
- \(x^2 - 5x - 14 = (x - 7)(x + 2)\)
- \(28 + 3x - x^2 = -(x^2 - 3x - 28) = -(x - 7)(x + 4)\)

So the expression becomes:
\[
\frac{(x - 7)(x + 2)}{-(x - 7)(x + 4)}
\]

#### Step 2: Simplify by canceling common factors
- The \(x - 7\) terms in the numerator and denominator cancel out.

This leaves:
\[
\frac{x + 2}{-(x + 4)} = -\frac{x + 2}{x + 4}
\]

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

---

Problem 6:

Simplify:
\[
\frac{x(x - 3)(x + 6)}{x^2 + x - 12}
\]

#### Step 1: Factorize the denominator
- \(x^2 + x - 12 = (x + 4)(x - 3)\)

So the expression becomes:
\[
\frac{x(x - 3)(x + 6)}{(x + 4)(x - 3)}
\]

#### Step 2: Simplify by canceling common factors
- The \(x - 3\) terms in the numerator and denominator cancel out.

This leaves:
\[
\frac{x(x + 6)}{x + 4}
\]

#### Final Answer:
\[
\boxed{\frac{x(x + 6)}{x + 4}}
\]

---

Problem 7:

Simplify:
\[
\frac{x^3 - 9x^2}{x^2 - 3x - 54}
\]

#### Step 1: Factorize each term
- \(x^3 - 9x^2 = x^2(x - 9)\)
- \(x^2 - 3x - 54 = (x - 9)(x + 6)\)

So the expression becomes:
\[
\frac{x^2(x - 9)}{(x - 9)(x + 6)}
\]

#### Step 2: Simplify by canceling common factors
- The \(x - 9\) terms in the numerator and denominator cancel out.

This leaves:
\[
\frac{x^2}{x + 6}
\]

#### Final Answer:
\[
\boxed{\frac{x^2}{x + 6}}
\]

---

Problem 8:

Simplify:
\[
\frac{x^2(x^2 + 3x + 2)}{2x(x - 4)(x + 2)}
\]

#### Step 1: Factorize each term
- \(x^2 + 3x + 2 = (x + 1)(x + 2)\)

So the expression becomes:
\[
\frac{x^2(x + 1)(x + 2)}{2x(x - 4)(x + 2)}
\]

#### Step 2: Simplify by canceling common factors
- The \(x\) terms in the numerator and denominator cancel out.
- The \(x + 2\) terms in the numerator and denominator cancel out.

This leaves:
\[
\frac{x(x + 1)}{2(x - 4)}
\]

#### Final Answer:
\[
\boxed{\frac{x(x + 1)}{2(x - 4)}}
\]

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Final Answers:

1. \(\boxed{\frac{3}{4}}\)
2. \(\boxed{\frac{3}{2(2x + 3)}}\)
3. \(\boxed{\frac{7(x + 5)}{2}}\)
4. \(\boxed{x^2}\)
5. \(\boxed{-\frac{x + 2}{x + 4}}\)
6. \(\boxed{\frac{x(x + 6)}{x + 4}}\)
7. \(\boxed{\frac{x^2}{x + 6}}\)
8. \(\boxed{\frac{x(x + 1)}{2(x - 4)}}\)