https://file.aiquickdraw.com/mathbot/file/1729471361102nt8riqy8.jpg

Let's examine the polynomial function provided in the question:

\[ f(x) = -x^2(x - 2)(x + 4) \]

The question asks you to determine the graph's end behavior using the **Leading Coefficient Test**.

### Key Points:
1. **Degree of the polynomial**:
   - We can expand \( f(x) \) to analyze its degree.
   - The highest power of \( x \) comes from multiplying \( x^2(x - 2)(x + 4) \), resulting in \( -x^4 \).
   - Thus, this is a 4th-degree polynomial.

2. **Leading coefficient**:
   - The leading term is \( -x^4 \).
   - The negative sign indicates the leading coefficient is negative.

### End Behavior Based on Degree and Leading Coefficient:
- **Even Degree (4)**: The ends of the graph go in the same direction.
- **Negative Leading Coefficient**: Since the leading coefficient is negative, the graph **falls** as it approaches both \( +\infty \) and \( -\infty \).

Thus, the correct end behavior is:
- The graph **falls to the left** and **falls to the right**.

### Answer:
- The correct choice is **A**: "The graph of \( f(x) \) falls to the left and falls to the right."

Would you like any additional details or clarifications?

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Here are five related questions:
1. How does the degree of a polynomial influence the graph's end behavior?
2. What happens to the graph of a polynomial when the leading coefficient is positive?
3. How can you determine the turning points of a polynomial function?
4. What is the significance of the sign of the leading coefficient in graphing polynomials?
5. Can the degree of a polynomial be odd, and how would that affect the end behavior?

**Tip**: Always check both the degree and the leading coefficient when using the Leading Coefficient Test!

Use the Leading Coefficient Test to determine the graph's end behavior for the polynomial function f(x) = -x^2(x - 2)(x + 4).

Learn how to determine the end behavior of the polynomial function f(x) = -x^2(x - 2)(x + 4) using the Leading Coefficient Test. This problem covers key concepts like the degree of the polynomial and the sign of the leading coefficient. Suitable for students in Grades 10-12.

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