From the graph, we can make several observations:

1. The graph crosses the x-axis at $x = 0$ and $x = -2$. These points represent the roots of the polynomial, and they appear to have even multiplicities because the curve touches but does not cross the x-axis at these points.
2. The behavior near $x = 0$ suggests the root at $x = 0$ has a higher multiplicity than that at $x = -2$, as the graph flattens out more near this point.
3. The graph dips between the roots, indicating a negative coefficient for the highest degree term.

Now let's analyze the answer choices:

• $p(x) = x^2(x + 2)^2$: This represents a polynomial with both roots $x = 0$ and $x = -2$ having multiplicity 2. This would result in a more symmetric graph around the x-axis, so this is likely not the correct answer.
• $p(x) = x^2(x + 2)^3$: The root at $x = 0$ has multiplicity 2, while the root at $x = -2$ has multiplicity 3. This might produce the behavior we see in the graph, as the root at $x = -2$ is less flat than at $x = 0$, but it doesn’t match the observation that the curve touches the x-axis at $x = -2$ without crossing.
• $p(x) = x^3(x + 2)^3$: Both roots would have odd multiplicities, meaning the curve would cross the x-axis at both points, which doesn't fit the graph.
• $p(x) = x^3(x + 2)^2$: The root at $x = 0$ has multiplicity 3, and the root at $x = -2$ has multiplicity 2. This matches the graph because the curve flattens at $x = 0$ (indicating higher multiplicity), and touches at $x = -2$ (indicating even multiplicity).

The correct answer is D: $p(x) = x^3(x + 2)^2$.

Would you like more details on the reasoning process or have any questions?

Here are 5 related questions for further exploration:

1. How does the multiplicity of a root affect the shape of the graph at that root?
2. What is the general form of a polynomial if you know its roots and their multiplicities?
3. How can you determine the leading term of a polynomial from its graph?
4. How does the end behavior of a polynomial help in identifying its degree?
5. What are the differences in graph behavior between even and odd multiplicity roots?

Tip: The degree of a polynomial is the sum of the multiplicities of its roots, and it dictates the end behavior of the graph.