The graph provided represents a function with specific roots and behavior. Let's analyze it in detail and match it with the correct function from the given options.

Step-by-Step Analysis:

1. Roots of the Function:

   • The graph crosses the x-axis at $x = -1$ and $x = 2$. These are the roots of the function.
   • From this, we can infer that the factors of the function must include $(x + 1)$ and $(x - 2)$.

2. Behavior near the Roots:

   • At $x = -1$, the graph passes straight through the x-axis, indicating a single root (linear factor).
   • At $x = 2$, the graph "bounces" off the x-axis, indicating a double root (squared factor).

3. Candidate Functions:

   • Now we look at the given options to find the function with a root at $x = -1$ (linear) and $x = 2$ (squared).

   Let's check the options:

   • Option A: $f(x) = (x + 1)(x - 2)^2$
     • This matches the graph: a root at $x = -1$ and a double root at $x = 2$.
   • Option B: $f(x) = (x - 1)(x + 2)^2$
     • This has roots at $x = 1$ and $x = -2$, which does not match the graph.
   • Option C: $f(x) = (x + 1)(x - 2)$
     • This does not have the correct behavior at $x = 2$ (should be squared).
   • Option D: $f(x) = (x - 2)(x + 1)^2$
     • This has the wrong root multiplicity at $x = -1$.

Conclusion:

The correct function is Option A: $f(x) = (x + 1)(x - 2)^2$.

Would you like further explanation or details? Here are some additional questions you can explore:

1. How do you determine the multiplicity of a root from a graph?
2. What would the graph look like for a function with a double root at both $x = -1$ and $x = 2$?
3. How do changes in the signs of factors affect the graph?
4. What would happen to the graph if one of the factors were cubed instead of squared?
5. How can the derivative of this function help confirm its turning points and behavior?

Tip: Always look at the behavior of the graph around the roots to understand whether they are single, double, or higher-order roots.