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?

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!