\(B^3A < 0\) and \(A > 0\)
Now we know that A is Positive and A * (odd power of B) < 0
=> B < 0
So, A > 0 and B < 0
Now let's look at the answer choices

A. AB will be Positive * Negative => Negative. So, A is the answer and we do not need to check further, but writing the solution to complete the problem.
B. \(B^2A\) => \((Negative)^2positve\) => Positive * Positive = Positive
C. \(B^4\) => \((Negative)^4\) => Positive
D. \(\frac{A}{B^2}\) => \(\frac{Positive}{(Negative)^2}\) => \(\frac{Positive}{Positive}\) => Positive
E. \(-\frac{B}{A}\) => \(-\frac{Negative}{Positive}\) => -(Negative) => Positive

So, answer will be A
Hope it helps!

Refer the following videos to know about inequalities in more detail