Not sure you're in the right sub-forum, but anyway: you first need to simplify!

f(x)=(4x^2-4x-8)/(x+1+x+1)

= 4(x^2-x-2) / 2(x+1)

= 4(x+1)(x-2) / 2(x+1)

= 2(x-2)

= 2x-4

See, so that funny-looking equation can in fact be simplified to a vanilla linear equation. You probably started worrying about parabolas but the key insight was that the quadratic trinomial (the numerator of the fraction) was divided by an expression that contains an x term, and that's a rather suspicious-looking expression at that. Often times, when you have things like x+1+x+1 it's a hint that you'll be able to factor out another (x+1) somewhere else.

So really, the question is "what is the range and domain of a linear function with a non-zero coefficient on the x (non-zero slope)"? The answer is all real numbers, since a line has an inifinite length! Note that if it had a zero coefficient on the x, your line would have a flat slope and f(x) would be equal to a constant (the y-intercept), so the domain would be all real numbers, but the range would be only that constant.

(disclaimer: I'm still in the middle of learning quadratics and stuff, so my answer should be taken with a pinch of salt. Someone will correct me if need be!)