The Intermediate Value Theorem states that on a closed interval [a,b], if there is a value, d, between f(a) and f(b), then there is a point, c, on the interval [a,b] such that f(c) = d.
So how does this relate to your problem? Simple. In your problem we want to know if there is a value f(x=c) = 0 on the interval. So compute f(a) and f(b) for each of the candidate intervals and see if f(x)=0 lies between them. If it does, it's the answer.
I'll do A (Between 0 and 1) for you:
f(0) = (0)^{4}-3(0)^{2}-8=-8
f(1) = (1)^{4}-3(1)^{2}-8 = 1-3-8 = -10
f(x) = 0 does not lie between f(0)=-8 and f(1)=-10, so choice A is not the answer.