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.

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.