Multiply as indicated.
StartFraction x squared minus 64 Over x squared minus 4 x minus 32 EndFraction times StartFraction x squared plus 12 x plus 32 Over x squared plus 4 x minus 32 EndFraction
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Part 1
StartFraction x squared minus 64 Over x squared minus 4 x minus 32 EndFraction times StartFraction x squared plus 12 x plus 32 Over x squared plus 4 x minus 32 EndFraction
equals
  
enter your response here ​(Simplify your​ answer.)

We are tasked with multiplying the following two rational expressions and simplifying the result:

\[
\frac{x^2 - 64}{x^2 - 4x - 32} \times \frac{x^2 + 12x + 32}{x^2 + 4x - 32}
\]

### Step 1: Factor each quadratic expression

1. **Factor \(x^2 - 64\)**:
   This is a difference of squares:
   \[
   x^2 - 64 = (x - 8)(x + 8)
   \]

2. **Factor \(x^2 - 4x - 32\)**:
   We need to find two numbers that multiply to \(-32\) and add to \(-4\). These numbers are \(-8\) and \(4\):
   \[
   x^2 - 4x - 32 = (x - 8)(x + 4)
   \]

3. **Factor \(x^2 + 12x + 32\)**:
   We need to find two numbers that multiply to \(32\) and add to \(12\). These numbers are \(4\) and \(8\):
   \[
   x^2 + 12x + 32 = (x + 4)(x + 8)
   \]

4. **Factor \(x^2 + 4x - 32\)**:
   We need to find two numbers that multiply to \(-32\) and add to \(4\). These numbers are \(8\) and \(-4\):
   \[
   x^2 + 4x - 32 = (x + 8)(x - 4)
   \]

### Step 2: Substitute the factored forms

Substituting the factored forms into the original expression:

\[
\frac{(x - 8)(x + 8)}{(x - 8)(x + 4)} \times \frac{(x + 4)(x + 8)}{(x + 8)(x - 4)}
\]

### Step 3: Simplify the expression

Now, cancel out common factors in the numerator and denominator:

- The factor \((x - 8)\) appears in the numerator and denominator, so it can be canceled.
- The factor \((x + 8)\) also appears in both, so it can be canceled.
- The factor \((x + 4)\) appears in both, so it can be canceled as well.

After canceling, the expression simplifies to:

\[
\frac{1}{x - 4}
\]

### Final Answer:
\[
\frac{1}{x - 4}
\]

Would you like to explore any specific details about the simplification process or have any other questions?

Here are 5 related questions to help deepen your understanding:
1. How do you factor a quadratic equation?
2. What is the difference of squares, and how does it help in factoring?
3. What is the general method for simplifying rational expressions?
4. How do you identify and cancel common factors in rational expressions?
5. How do you handle situations where expressions cannot be factored easily?

**Tip:** Always look for common factors in both the numerator and denominator before multiplying rational expressions. This can greatly simplify the work!

Multiply as indicated. StartFraction x squared minus 64 Over x squared minus 4 x minus 32 EndFraction times StartFraction x squared plus 12 x plus 32 Over x squared plus 4 x minus 32 EndFraction equals (Simplify your answer.)

Learn how to multiply and simplify rational expressions by factoring quadratic equations. This step-by-step solution covers key algebraic techniques like the difference of squares and factoring trinomials, suitable for Grades 9-10.

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