https://file.aiquickdraw.com/mathbot/file/1729519847838xsk8q2no.jpg

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?

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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.

Which graph is a possible sketch of the function g(x) = (x + 1)^2(x - 2)?

Determine which graph correctly represents the polynomial function g(x) = (x + 1)^2(x - 2). This problem involves analyzing roots, multiplicity, and end behavior of the function. Suitable for high school students in Grades 10-12.

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