The question asks about the graph of the function $h(x)$ and requires information about the roots, their multiplicities, and the degree of the polynomial $h(x)$. Let’s analyze the graph in detail:

Observations from the graph:

1. Root at $x = -1$:

   • The graph touches the x-axis but does not cross it at $x = -1$, which suggests a root of even multiplicity (likely $2$).

2. Root at $x = 2$:

   • The graph crosses the x-axis at $x = 2$, and the crossing indicates a root of odd multiplicity (likely $1$).

3. Root at $x = 3$:

   • The graph also crosses the x-axis at $x = 3$, with a behavior similar to $x = 2$, suggesting a root of odd multiplicity (likely $1$).

4. Degree of the polynomial:

   • The graph’s end behavior shows that as $x \to -\infty$, $h(x) \to \infty$, and as $x \to \infty$, $h(x) \to -\infty$.
   • This indicates that $h(x)$ is a polynomial of odd degree (likely $5$, as it has three distinct roots, two of which are simple and one with higher multiplicity).

Completing the blanks:

1. Multiplicity of root at $x = -1$: $2$ (even multiplicity as the graph touches the axis but does not cross it).
2. Multiplicity of root at $x = 2$: $1$ (odd multiplicity as the graph crosses the axis).
3. Multiplicity of root at $x = 3$: $1$ (odd multiplicity as the graph crosses the axis).
4. Degree of $h(x)$: $5$ (based on the end behavior and the roots).

Would you like me to elaborate further on how to determine the degree or multiplicities?

Here are 5 related questions to explore further:

1. What happens to the graph of a polynomial function at a root with multiplicity 3 or higher?
2. How does end behavior vary with the leading coefficient and degree of a polynomial?
3. Can a polynomial of degree 5 have more than three distinct real roots? Why or why not?
4. How is the degree of a polynomial related to the number of turning points on its graph?
5. What are the differences in the graph’s shape at roots with multiplicities 1, 2, or 3?

Tip: Always check the end behavior of a polynomial function to determine whether its degree is odd or even.