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The given polynomial function is \( P(x) = (x - 3)(x - 2)(x + 1) \), and the domain is \( -5 \leq x \leq 5 \). We are asked to determine which statement about the graph of \( P(x) \) is true.

### Step 1: Analyze the roots of \( P(x) \)
The roots are found by setting \( P(x) = 0 \):
\[
(x - 3)(x - 2)(x + 1) = 0
\]
The roots are \( x = 3 \), \( x = 2 \), and \( x = -1 \). So, the graph intersects the x-axis at these three points.

### Step 2: Determine the degree and general shape of the polynomial
The polynomial \( P(x) \) is a cubic function (degree 3), and the leading term is positive since the expansion would give \( x^3 \). A cubic function generally has:
- One or two turning points (local extrema).
- One global extrema on the interval if the function has endpoints that define maximum or minimum values.

### Step 3: Local extrema and global extrema
To find the critical points (where local extrema may occur), we would differentiate \( P(x) \) and find where the derivative is zero. However, based on the shape of cubic functions, there are generally two turning points corresponding to local extrema. For global extrema, since the function is defined on a closed interval \( [-5, 5] \), there will be global extrema as well. 

### Step 4: Evaluate the statements
- **Option A** mentions four local extrema, which is not possible for a cubic function (which can only have up to two turning points).
- **Option B** correctly states that there are two local extrema that are not global extrema, and there are two global extrema. This aligns with the typical behavior of a cubic polynomial on a closed interval.
- **Option C** incorrectly states there are no local extrema, which is false for a cubic function.
- **Option D** states there are no global extrema, which is false since the function is defined on a closed interval, meaning it must have global extrema.

### Conclusion:
The correct answer is **B**.

Would you like further clarification on how to derive extrema or the shape of the function? 

Here are 5 related questions to explore:
1. How do you find the critical points of a cubic polynomial?
2. What is the difference between local and global extrema?
3. How do you determine whether a turning point is a maximum or a minimum?
4. Why must a continuous function on a closed interval have a global maximum and minimum?
5. How does the degree of a polynomial affect the number of extrema?

**Tip:** Always check the behavior of a polynomial at its endpoints when it's defined over a closed interval, as this helps identify global extrema.

The polynomial function P is given by P(x) = (x - 3)(x - 2)(x + 1), and the domain of P is -5 ≤ x ≤ 5. Which of the following is true about the graph of P?

Learn how to analyze the cubic polynomial function P(x) = (x - 3)(x - 2)(x + 1) over the domain -5 ≤ x ≤ 5. This problem covers key concepts such as local extrema, global extrema, and root analysis, suitable for students in Grades 11-12.

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