Answer
$$\frac{f(x+h) - f(x)}{h}=-2x-h$$
Work Step by Step
If $f(x) = 4 - x^{2}$, then finding $\frac{f(x+h) - f(x)}{h}$ boils down to finding $f(x + h)$:
$$f(x + h) = 4 - (x + h)^{2}$$
$$f(x+h) = 4 - (x^{2} + 2xh + h^{2})$$
and substituting in the original function:
$$\frac{f(x+h) - f(x)}{h} = \frac{[4 - (x^{2} + 2xh + h^{2})] - [4 - (x^{2})]}{h}$$
$$= \frac{4 - x^{2} - 2xh - h^{2} - 4 + x^{2}}{h}$$
$$= \frac{-2xh -h^{2}}{h}$$
$$= \frac{h(-2x -h)}{h}$$
$$=-2x-h$$
