https://file.aiquickdraw.com/mathbot/file/1726848098994in5kylrl.jpg

The polynomial function provided in the image is:

\[
f(x) = -2x^5(6x - 9)(x^2 + 1)^2
\]

We are tasked with determining the correct behavior of the function \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \).

### Step-by-step analysis:

1. **Degree of the polynomial**:
   - The term \( -2x^5 \) dominates the behavior of the polynomial as \( x \to \infty \) or \( x \to -\infty \).
   - The degree of \( f(x) \) is determined by \( x^5 \), and since it is negative (because of the factor \(-2\)), the function will behave similarly to a negative leading term for large values of \( |x| \).

2. **As \( x \to \infty \)**:
   - As \( x \to \infty \), the leading term \( -2x^5 \) will dominate, and since the coefficient of the highest power of \( x \) is negative, the limit of \( f(x) \) as \( x \to \infty \) will be:
   \[
   \lim_{x \to \infty} f(x) = -\infty
   \]

3. **As \( x \to -\infty \)**:
   - As \( x \to -\infty \), the leading term \( -2x^5 \) will also dominate, and since \( x^5 \) for negative \( x \) will still be negative (odd power of a negative number), the product with \(-2\) will make it positive. Thus, the limit of \( f(x) \) as \( x \to -\infty \) will be:
   \[
   \lim_{x \to -\infty} f(x) = \infty
   \]

### Conclusion:
The correct answer is:

\[
\boxed{\lim_{x \to \infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = \infty}
\]

This corresponds to the second option in the image, which is highlighted as the correct choice.

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Would you like further details on this solution or have any questions? Here are 5 related questions you can explore:

1. What is the general behavior of polynomials based on their leading term?
2. How does the degree of a polynomial affect its end behavior?
3. Why do odd and even powers of \( x \) behave differently as \( x \to \infty \)?
4. What role do the other factors of the polynomial (e.g., \( (6x - 9) \) and \( (x^2 + 1)^2 \)) play in its overall behavior?
5. How can you graphically interpret the limits at infinity for polynomials?

**Tip**: Always pay close attention to the sign and degree of the leading term in a polynomial when determining its end behavior.

The polynomial is given by f(x) = -2x^5(6x - 9)(x^2 + 1)^2. Which of the following is a true statement about the function f?

Analyze the end behavior of the polynomial f(x) = -2x^5(6x - 9)(x^2 + 1)^2 by studying its limits as x approaches infinity and negative infinity. Learn how to apply leading term analysis and the Polynomial End Behavior Theorem.

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