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To find the derivative of the function \( h(z) = (z^3 - z^2 + 4z)(z - 6) \) using the **Product Rule**, we apply the following formula:

\[
\frac{d}{dz}[f(z)g(z)] = f'(z)g(z) + f(z)g'(z)
\]

Here:
- \( f(z) = z^3 - z^2 + 4z \)
- \( g(z) = z - 6 \)

### Step 1: Compute \( f'(z) \) and \( g'(z) \)
- \( f'(z) = 3z^2 - 2z + 4 \)
- \( g'(z) = 1 \) (since the derivative of \( z \) is 1 and the derivative of \( -6 \) is 0)

### Step 2: Apply the Product Rule
\[
h'(z) = f'(z)g(z) + f(z)g'(z)
\]

Substitute the expressions for \( f(z) \), \( g(z) \), \( f'(z) \), and \( g'(z) \):

\[
h'(z) = (3z^2 - 2z + 4)(z - 6) + (z^3 - z^2 + 4z)(1)
\]

### Step 3: Simplify
Now, expand the terms:
\[
(3z^2 - 2z + 4)(z - 6) = 3z^3 - 18z^2 - 2z^2 + 12z + 4z - 24 = 3z^3 - 20z^2 + 16z - 24
\]
\[
(z^3 - z^2 + 4z)(1) = z^3 - z^2 + 4z
\]

Now, sum the two results:
\[
h'(z) = (3z^3 - 20z^2 + 16z - 24) + (z^3 - z^2 + 4z)
\]
\[
h'(z) = 4z^3 - 21z^2 + 20z - 24
\]

### Conclusion:
The correct option is **D**, which matches the expanded form:
\[
h'(z) = (z - 6)(3z^2 - 2z + 4) + (z^3 - z^2 + 4z)
\]

Do you need more details, or any further explanation?

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Here are 5 related questions for practice:
1. What is the derivative of \( h(z) = (z^4 - z^3)(z + 5) \) using the product rule?
2. How do you apply the product rule when there are more than two functions being multiplied?
3. What happens when one of the functions in a product rule is a constant?
4. Can you derive \( h(z) = (e^z)(\sin z) \) using the product rule?
5. How can you verify a derivative by expanding the original product first?

**Tip:** Always simplify after applying the product rule to make it easier to combine terms.

Find the derivative of h(z) = (z^3 - z^2 + 4z)(z - 6) using the Product Rule. Then verify by expanding the product.

This problem demonstrates how to find the derivative of the function h(z) = (z^3 - z^2 + 4z)(z - 6) using the product rule. Follow a detailed step-by-step process, applying the product rule and verifying the result by expanding the product.

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