Let's analyze the given polynomial graph and equation options.

Step 1: Identify the roots (x-intercepts)

From the graph:

• The polynomial has x-intercepts at $x = -2$ and $x = 2.5$.
• There's a "bounce" at $x = 2.5$, indicating a double root.

Thus, the factors associated with these roots should be:

• For $x = -2$, the factor is $(x + 2)$.
• For $x = 2.5$ or $x = 5/2$, the factor is $(2x - 5)$, and since this is a double root, it should be squared: $(2x - 5)^2$.

Step 2: Analyze the options

Now, look at each option to see which matches the structure derived from the graph:

• Option a: $p(x) = x^2(2x - 5)(x + 2)$

  • This does not match because $(2x - 5)$ is not squared, but the graph shows a double root at $x = 2.5$.

• Option b: $p(x) = x^2(2x - 5)(x + 2)^2$

  • This suggests a double root at $x = -2$, but the graph shows a single root at $x = -2$, so this is incorrect.

• Option c: $p(x) = x^2(2x - 5)^2(x + 2)$

  • This fits the structure of the graph: single root at $x = -2$, double root at $x = 2.5$, and matches the squaring of $(2x - 5)$.

• Option d: $p(x) = x(2x - 5)^2(x + 2)$

  • This is close, but there is no additional $x^2$ factor for proper degree.

Step 3: Conclusion

The correct equation that matches the graph is option c, $p(x) = x^2(2x - 5)^2(x + 2)$.

Would you like more details on the steps?

Here are five related questions to expand on this:

1. What does it mean when a root "bounces" on the x-axis in a polynomial graph?
2. How do we determine the degree of a polynomial from its graph?
3. What is the significance of multiplicity of roots in polynomials?
4. How do you expand a polynomial like $(2x - 5)^2(x + 2)$?
5. Can a polynomial have both real and imaginary roots?

Tip: When a root touches the x-axis but does not cross, it has even multiplicity, typically 2 for a bounce effect.