Show that the function f(x)= x^(3) +3/(x^2) +2 has exactly one zero on the interval (-infinity, 0).

Question

So far this is what I have:

0=x^3 + 3/(x^2) +2
-2= (1/x^2)(x^5 + 3)
-2x^2= x^5 +3

But now I'm stuck. I also am not sure if this is how I'm supposed to be solving the problem. We had been learning Rolle's Theorem and the Mean Value Theorem, but none of those seem applicable here because they require that all points on the interval be differentiable, and if the function has a zero on the interval then it is not differentiable at that point. Am I going about this the right way? And how would I solve it from here?