10. which graph is of the polynomial y = -(x + 1)(x - 2)^2

Explanation:

Step1: Find the roots

Set $y = 0$. Then $-(x + 1)(x - 2)^2=0$. By the zero - product property, $x+1 = 0$ gives $x=-1$ and $(x - 2)^2=0$ gives $x = 2$. So the roots are $x=-1$ and $x = 2$.

Step2: Analyze the multiplicity

The factor $(x + 1)$ has multiplicity 1, so the graph crosses the x - axis at $x=-1$. The factor $(x - 2)$ has multiplicity 2, so the graph touches the x - axis at $x = 2$.

Step3: Determine the end - behavior

The leading term of the polynomial $y=-(x + 1)(x - 2)^2=-(x + 1)(x^{2}-4x + 4)=-x^{3}+3x^{2}-4$ is $-x^{3}$. Since the leading coefficient is negative and the degree is 3 (odd), as $x\to-\infty$, $y\to\infty$ and as $x\to\infty$, $y\to-\infty$.

Answer:

The graph that crosses the x - axis at $x=-1$, touches the x - axis at $x = 2$ and has the correct end - behavior (rises to the left and falls to the right) is the correct one. Without seeing the full set of options clearly, based on the described characteristics, you can identify the right graph among the choices.