Let's analyze the given polynomial function $P(x) = -x^5 - 5x^3 - 5x + 2$ to determine its end behavior and select the correct end behavior diagram.

Key information:

1. Degree of the polynomial: The highest power of $x$ in the given polynomial is 5, meaning the polynomial has degree 5.

2. Leading coefficient: The coefficient of the highest degree term, $-x^5$, is negative.

End behavior rules for polynomials:

• Odd-degree polynomials (like $x^5$) have opposite behaviors at $x \to +\infty$ and $x \to -\infty$.
• Negative leading coefficient: If the leading coefficient is negative, the polynomial will fall to $-\infty$ as $x \to +\infty$ and rise to $+\infty$ as $x \to -\infty$.

End behavior of $P(x)$:

• As $x \to +\infty$, $P(x) \to -\infty$.
• As $x \to -\infty$, $P(x) \to +\infty$.

Conclusion:

The end behavior diagram should show the function rising on the left (as $x$ goes to negative infinity) and falling on the right (as $x$ goes to positive infinity).

From the image, Option C corresponds to this behavior, with the left arrow pointing upward and the right arrow pointing downward.

Would you like more details, or do you have any other questions?

Here are 5 related questions for further exploration:

1. What are the general rules for determining end behavior based on degree and leading coefficient?
2. How does the sign of the leading coefficient affect the shape of a polynomial function?
3. Can a higher-degree term dominate the end behavior if it has a smaller coefficient?
4. What would the end behavior be for a polynomial of degree 6 with a positive leading coefficient?
5. How does graphing a polynomial help in visualizing its end behavior?

Tip: Always check the degree and leading coefficient first to quickly determine end behavior of polynomials.