The fact that "lim f(x) = - infinty as x-> -4" and "lim f(x) = - infinty as x-> 2-" tells you that there must be factors of (x+ 4) and (x- 2) in the denominator. Further, since the limit, as x goes to 4 from "both sides" is there must be a factor of in the denominator. That means that the degree of the denominator is at least 3. In order that the limits as x goes to and be non-zero real numbers, the degree of the numerator and denominator must be the same. I suggest you write the function as
and try to determine values for a, b, c, and d that make the other requirements true.