1. The problem asks to describe the end behavior of a polynomial function $P$ which is the sum of two polynomials: one of degree 2 with a positive leading coefficient, and one of degree 3 with a negative leading coefficient. 2. Recall that the end behavior of a polynomial is dominated by the term with the highest degree because as $x$ becomes very large or very small, the highest degree term grows faster than the others. 3. Here, the polynomial of degree 3 has a negative leading coefficient, and the polynomial of degree 2 has a positive leading coefficient. Since degree 3 is higher than degree 2, the degree 3 term dominates the end behavior. 4. For a cubic polynomial with a negative leading coefficient, as $x \to \infty$, $y \to -\infty$, and as $x \to -\infty$, $y \to \infty$. 5. Let's write example polynomials: $$f(x) = -x^3 + 2x^2$$ Here, $-x^3$ is degree 3 with negative leading coefficient, and $2x^2$ is degree 2 with positive leading coefficient. 6. Their sum $P(x) = f(x) = -x^3 + 2x^2$. 7. As $x \to \infty$, the $-x^3$ term dominates, so $P(x) \to -\infty$. 8. As $x \to -\infty$, the $-x^3$ term dominates, and since $(-x)^3 = -x^3$, $-x^3$ becomes positive large, so $P(x) \to \infty$. 9. Therefore, the end behavior matches option A: As $x \to \infty$, $y \to -\infty$, and as $x \to -\infty$, $y \to \infty$. Final answer: A