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The problem asks you to multiply two rational expressions and simplify them. The expressions are:

\[
\frac{z^2 + 64}{z^2 - 16} \times \frac{z^2 + 8z + 16}{z^2 - 4z + 16}
\]

### Step 1: Factor each term

1. **For the numerator \(z^2 + 64\)**:
   - This is a sum of squares, which does not factor nicely over real numbers.

2. **For the denominator \(z^2 - 16\)**:
   - This is a difference of squares:
   \[
   z^2 - 16 = (z - 4)(z + 4)
   \]

3. **For the numerator \(z^2 + 8z + 16\)**:
   - This is a perfect square trinomial:
   \[
   z^2 + 8z + 16 = (z + 4)^2
   \]

4. **For the denominator \(z^2 - 4z + 16\)**:
   - This quadratic does not factor easily, so we leave it as is.

### Step 2: Multiply the factored forms

Now we multiply the factored forms:

\[
\frac{z^2 + 64}{(z - 4)(z + 4)} \times \frac{(z + 4)^2}{z^2 - 4z + 16}
\]

### Step 3: Simplify the expression

We can cancel out common terms:

- The factor \(z + 4\) appears both in the numerator and denominator, so we can cancel one \(z + 4\) from both sides.

Thus, the expression simplifies to:

\[
\frac{z^2 + 64}{(z - 4)(z^2 - 4z + 16)}
\]

### Final Answer:

- **Numerator**: \(z^2 + 64\)
- **Denominator**: \((z - 4)(z^2 - 4z + 16)\)

Would you like more details on any part of the solution? Here are five related questions to explore:

1. How do you factor a perfect square trinomial?
2. What is the difference between a sum of squares and a difference of squares?
3. Why can't we factor certain quadratic expressions easily?
4. Can you explain why some terms cancel out when multiplying rational expressions?
5. How would the process change if the expressions were subtracted instead of multiplied?

**Tip:** When simplifying rational expressions, always look for common factors in the numerator and denominator to simplify early in the process!

Multiply the rational expressions and simplify: (z^2 + 64)/(z^2 - 16) × (z^2 + 8z + 16)/(z^2 - 4z + 16).

Learn how to multiply and simplify rational expressions such as (z^2 + 64)/(z^2 - 16) × (z^2 + 8z + 16)/(z^2 - 4z + 16). This problem demonstrates key algebraic concepts like factoring, difference of squares, and simplification of rational expressions. Suitable for students in Grades 9-12.

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