We are tasked with identifying the correct sketch for the function $g(x) = (x+1)^2(x-2)$.

Step-by-step analysis:

1. Roots of the Function: The function $g(x) = (x+1)^2(x-2)$ has roots where $g(x) = 0$.

   • $(x+1)^2 = 0$ gives $x = -1$ (a repeated root, so it will touch the x-axis here).
   • $(x-2) = 0$ gives $x = 2$ (a simple root, so it will cross the x-axis here).

2. Behavior at Roots:

   • At $x = -1$, the factor $(x+1)^2$ is squared, which means the graph will touch the x-axis but not cross it.
   • At $x = 2$, the factor $(x-2)$ is linear, so the graph will cross the x-axis.

3. End Behavior:

   • As $x \to -\infty$, $g(x) \to -\infty$, because the highest degree term (when you expand the function) will be negative.
   • As $x \to \infty$, $g(x) \to \infty$, indicating the graph will rise on the right side.

Conclusion:

• The graph must touch the x-axis at $x = -1$ and cross the x-axis at $x = 2$.
• Based on these characteristics, the correct sketch is graph d, as it displays the appropriate behavior at the roots and the correct end behavior.

Would you like more details on the graph's properties or behavior?

Here are 5 related questions to explore further:

1. How do you determine the end behavior of a polynomial function?
2. What is the significance of a double root in a polynomial?
3. How does the degree of each factor influence the shape of the graph?
4. How can you find turning points and concavity from the derivative of a function?
5. How does the sign of the leading coefficient affect the graph's orientation?

Tip: Always analyze the roots and multiplicities of polynomial factors to predict graph behavior at key points.