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The problem in the image involves analyzing the polynomial function \( f(x) = x^2(x - 2)^3(x + 3) \) and answering several questions regarding its graph and x-intercepts.

### a) End behavior of the graph using the Leading Coefficient Test:
The given polynomial function is \( f(x) = x^2(x - 2)^3(x + 3) \).

1. The degree of the polynomial can be determined by adding the exponents:
   - \( x^2 \) has a degree of 2,
   - \( (x - 2)^3 \) has a degree of 3,
   - \( (x + 3) \) has a degree of 1.

   So the total degree of the polynomial is \( 2 + 3 + 1 = 6 \), which is even.

2. The leading term, which dominates the end behavior, comes from multiplying the highest degree terms from each factor:
   - The leading term is \( x^6 \), which has a positive coefficient.

For even degree polynomials with a positive leading coefficient, the end behavior is that the graph **rises to the left and rises to the right**.

Thus, option **C (The graph rises to the left and to the right)** is the correct choice.

### b) X-intercepts of the graph:
The x-intercepts are found by setting \( f(x) = 0 \). From the factored form \( f(x) = x^2(x - 2)^3(x + 3) \), the x-intercepts occur when each factor equals zero:

- \( x^2 = 0 \) gives \( x = 0 \),
- \( (x - 2)^3 = 0 \) gives \( x = 2 \),
- \( (x + 3) = 0 \) gives \( x = -3 \).

Thus, the x-intercepts are \( 0 \), \( 2 \), and \( -3 \).

### c) Where the graph crosses the x-axis:
To determine where the graph crosses the x-axis, we need to check the multiplicity of the roots:

- \( x = 0 \) has a multiplicity of 2 (from \( x^2 \)),
- \( x = 2 \) has a multiplicity of 3 (from \( (x - 2)^3 \)),
- \( x = -3 \) has a multiplicity of 1 (from \( (x + 3) \)).

A root with an odd multiplicity means the graph **crosses** the x-axis at that point, while a root with even multiplicity means the graph **touches** the x-axis and turns around.

- The graph **touches** the x-axis at \( x = 0 \) (even multiplicity),
- The graph **crosses** the x-axis at \( x = 2 \) and \( x = -3 \) (odd multiplicities).

Thus, the correct answer is:
- The graph crosses the x-axis at \( x = -3 \) and \( x = 2 \).

Let me know if you need further clarifications! 

### Relative Questions:
1. What does the degree of a polynomial tell you about its end behavior?
2. How do you determine the multiplicity of a root from the factored form of a polynomial?
3. Why does the graph only cross the x-axis at certain points?
4. How can the Leading Coefficient Test help predict the shape of the graph?
5. What are the implications of a root having even multiplicity on the behavior of the graph?

### Tip:
When analyzing polynomial graphs, always check the degree and leading coefficient first to predict end behavior and investigate root multiplicities to understand how the graph interacts with the x-axis.

For the polynomial function f(x) = x^2(x - 2)^3(x + 3), determine the graph's end behavior and find the x-intercepts.

Learn how to analyze the polynomial function f(x) = x^2(x - 2)^3(x + 3) by determining its end behavior and identifying the x-intercepts. This solution covers key algebraic concepts such as the Leading Coefficient Test, root multiplicity, and polynomial graph behavior, suitable for students in Grades 10-12.

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