Let's analyze the graph and the problem.

The graph of $p(x)$ has the following characteristics:

1. Roots: The graph crosses the x-axis at $x = -3$ and $x = 2$. This means these values are roots of the polynomial $p(x)$.
2. Behavior near the roots:
   • At $x = -3$, the graph touches the x-axis and bounces off. This indicates that $x = -3$ is a root with even multiplicity (likely a square, $(x + 3)^2$).
   • At $x = 2$, the graph also touches the x-axis and bounces off. This indicates that $x = 2$ is a root with even multiplicity (likely $(x - 2)^2$).

Now, let’s analyze the options:

• A. $(x - 3)(x + 2)$: The roots would be $x = 3$ and $x = -2$, which do not match the graph.
• B. $(x + 3)(x - 2)$: The roots would be $x = -3$ and $x = 2$, but the graph shows even multiplicities for these roots, and this option only provides multiplicity 1.
• C. $(x - 3)^2(x + 2)^2$: The roots are $x = -3$ and $x = 2$, both with even multiplicities. This matches the behavior of the graph.
• D. $(x + 3)^2(x - 2)^2$: The roots are $x = -3$ and $x = 2$, both with even multiplicities. However, the factors here are reversed from what is needed to match the graph.

Thus, the correct answer is C. $(x - 3)^2(x + 2)^2$.

Would you like a further breakdown or have any questions?

Here are 5 related questions to consider:

1. How do you determine the multiplicity of roots from a graph?
2. What happens to the graph of a polynomial at roots with odd versus even multiplicities?
3. Can the degree of a polynomial be determined from its graph?
4. How would the graph change if one of the roots had multiplicity 3 instead of 2?
5. How do you construct a polynomial if you are given its graph?

Tip: Always analyze the behavior of the graph at its roots to determine the multiplicity of each factor in the polynomial.