Hi, there. I'm happy to help with this.

First of all, on this DS question, more than half the battle is just simplifying the prompt. We have:

(p^2*q*r^3*s + p^2*q*r*s^3)/( p^2*q*r*s)
= (p^2*q*r^3*s)/( p^2*q*r*s) + (p^2*q*r*s^3)/( p^2*q*r*s)
= r^2 + s^2

So, really, the prompt just boils down to: is (r^2 + s^2) divisible by 8? A much much simpler question than the original! Among other things, p & q are now completely irrelevant to the question.

Statement #1: Each of p, q, r, and s are divisible by 2 but not by 4.

If r is divisible by 2 and not by 4, it has only one factor of two; in other words, r = 2*(odd number). The same is true of s. When squared, it has two and only two factors of 2; it is of the form r^2 = 4*(odd). Similarly, s^2 = 4*(odd). Then, we have r^2 + s^2 = 4*(odd) + 4*(odd) = 4*(odd + odd) = 4*(even), and 4 times any even number is always divisible by 8. This statement leads to a definitive yes answer, so it is sufficient.

Statement #2: Each of p, q, and r are divisible by 2 but not by 4, and s is an odd number.
For r, the same analysis as last time: r^2 = 4*(odd) = (even). If s = (odd), then s^2 = (odd). Then r^2 + s^2 = (even) + (odd) = (odd), and no odd number is divisible by 8. This statement leads to a definitive no answer, so it is sufficient.

Both statements by themselves sufficient. Answer = D.

Notice, incidentally, by official GMAT standards, this is a poor DS question, because the two statements contradict each other: they lead to opposite and contradictory answers to the prompt. That does not happen on the GMAT. By design, the information in one statement never contradicts the information in the other statement. That is a guaranteed feature of official GMAT Data Sufficiency.

Does all of that make sense? Please let me know if you have any questions about what I've said here.

Mike