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!