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.