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We are given a polynomial function \( f(x) = (2x + 5)(x + 3)(3x - 10)(2x - 8) \), which has zeros at \( x = -3 \), \( x = -\frac{5}{2} \), \( x = \frac{10}{3} \), and \( x = 4 \). The problem asks to determine the sign of \( f(x) \) over the interval \( -\frac{5}{2} < x < 4 \).

### Step-by-Step Breakdown:

1. **Zeros of the Function**: The zeros split the real line into intervals where the sign of the function can change. The zeros given are:
   - \( x = -3 \)
   - \( x = -\frac{5}{2} \)
   - \( x = \frac{10}{3} \)
   - \( x = 4 \)

2. **Interval to Analyze**: The interval given is \( -\frac{5}{2} < x < 4 \). We need to analyze the sign of the function in this interval, which is between \( x = -\frac{5}{2} \) and \( x = 4 \), specifically between the roots.

3. **Sign Analysis**:
   To find the sign of the function on this interval, we examine the sign of each factor in \( f(x) \) at representative points within subintervals. Specifically, for values of \( x \) between the zeros, we will pick points and determine if \( f(x) \) is positive or negative.

   - Choose a value between \( -\frac{5}{2} \) and \( \frac{10}{3} \) (e.g., \( x = 0 \)).
   - Also, check the sign of \( f(x) \) between \( \frac{10}{3} \) and \( 4 \) (e.g., \( x = 3.5 \)).

4. **Conclusion**:
   The function can change sign as we cross the roots. This means that within the interval \( -\frac{5}{2} < x < 4 \), the function will be sometimes positive and sometimes negative. Therefore, the correct answer is **C: \( f(x) \) is sometimes positive and sometimes negative on the interval**.

Would you like a deeper breakdown of how to analyze the sign changes or have any specific questions?

### Here are five related questions to expand on this topic:
1. How can you analyze the sign of a polynomial on a given interval more generally?
2. What is the graphical interpretation of the zeros of a polynomial?
3. How does the multiplicity of a root affect the behavior of the graph at that point?
4. How can you quickly sketch a polynomial function based on its factored form?
5. What is the role of the leading coefficient in determining the end behavior of the polynomial?

**Tip**: When analyzing the sign of a polynomial function, always check points between the zeros to understand how the function behaves across different intervals.

What is the sign of f(x) on the interval -5/2 < x < 4?

Learn how to determine the sign of a polynomial function f(x) = (2x + 5)(x + 3)(3x - 10)(2x - 8) over the interval -5/2 < x < 4. This solution covers key concepts like polynomial zeros, sign changes, and interval analysis, suitable for high school students in Grades 10-12.

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