Do you mean (x+2)(x-2)/x^2+4 or (x+2)(x-2)/(x^2+4)? The difference is important.
Written the first way, as you did, the function F cannot be evaluated at 0, for otherwise you get division by 0 (very bad). Any other number may be safely used for x, however. So the domain of F is the set of all nonzero numbers.
Written the second way, the function F is defined everywhere except x=2i and x=-2i, and hence the domain is the set of all complex numbers excluding 2i and -2i.