To solve the question, we need to analyze the graph and determine the correct polynomial function $p(x)$.

Step-by-step Analysis:

1. Roots/Zeros of the Polynomial: The graph crosses the x-axis at $x = -3$ and $x = 2$, which means the polynomial must have factors of $(x + 3)$ and $(x - 2)$.

2. Multiplicity of Roots:

   • The graph touches the x-axis at $x = -3$ and bounces off at $x = 2$. This indicates that:
     • At $x = -3$, the root has even multiplicity (since it just touches the x-axis).
     • At $x = 2$, the root has even multiplicity as well (since the graph bounces off the x-axis).

3. Degree of the Polynomial: Based on the shape of the graph (with two turning points), the degree of the polynomial seems to be at least 4.

Matching the Polynomial with the Given Options:

• From the analysis, the factors are likely $(x + 3)^2$ and $(x - 2)^2$ because both roots have even multiplicity.

Thus, the correct polynomial expression is:

$(x + 3)^2(x - 2)^2$

Answer:

The correct option is the third one: $(x + 3)^2(x - 2)^2$.

Would you like further details or explanations on any part of this? Here are some related questions to explore:

1. What is the general form of a polynomial with given roots?
2. How do multiplicities of roots affect the graph of a polynomial?
3. What is the significance of turning points in determining the degree of a polynomial?
4. How can you factor a polynomial given its graph?
5. What are the characteristics of polynomial graphs with even and odd multiplicity?

Tip: Remember that an even multiplicity causes the graph to touch or bounce off the x-axis, while an odd multiplicity causes it to cross.