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The given function is:

\[
f(x) = (2x + 5)(x + 3)(3x - 10)(2x - 8)
\]

The zeros of this function are at:

\[
x = -3, \quad x = -\frac{5}{2}, \quad x = \frac{10}{3}, \quad x = 4
\]

The question asks for the **sign** of the function \( f(x) \) on the interval:

\[
-\frac{5}{2} < x < 4
\]

### Steps to determine the sign of the function:

1. **Identify intervals based on zeros**: The zeros of the function split the real line into intervals. In this case, the relevant zeros are \( x = -\frac{5}{2} \) and \( x = 4 \). So, we examine the behavior of the function within these intervals:
   - \( \left(-\frac{5}{2}, 4\right) \)

2. **Test the sign of the function in the intervals**:
   - Pick test points within the interval to evaluate the sign of \( f(x) \). For example, choose \( x = 0 \), which is between \( -\frac{5}{2} \) and \( 4 \).
   - Evaluate \( f(0) = (2(0) + 5)(0 + 3)(3(0) - 10)(2(0) - 8) \)
   - \( f(0) = (5)(3)(-10)(-8) = 5 \times 3 \times (-10) \times (-8) = 1200 \)

   So, \( f(0) > 0 \), meaning \( f(x) \) is positive in this region.

3. **Behavior near the endpoints**:
   - Near \( x = -\frac{5}{2} \), \( f(x) \) changes sign, as this is a zero of the function.

### Conclusion:
Since the function changes sign at the zeros, the function \( f(x) \) is **sometimes positive and sometimes negative** within the interval \( -\frac{5}{2} < x < 4 \).

Thus, the correct answer is:

**C. \( f(x) \) is sometimes positive and sometimes negative on the interval.**

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Would you like more details about how we analyze signs of polynomials?

Here are some related questions for further exploration:

1. How do you determine the sign of a polynomial over an interval?
2. What happens to the sign of a function at its roots?
3. How does the multiplicity of a root affect the sign of a polynomial?
4. Can you describe how to sketch a polynomial function based on its factors and roots?
5. How do we handle polynomials with more complex factorizations?

**Tip**: When determining the sign of a function over an interval, always test points between the roots to ensure accuracy.

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

Determine the sign of the polynomial function f(x) = (2x + 5)(x + 3)(3x - 10)(2x - 8) on the interval -5/2 < x < 4. This problem involves analyzing zeros and intervals and is suitable for students in Grades 10-12.

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