Solve the difference quotient 
So I tried a variety of answers and they all came up incorrect. I guess I don't understand difference quotient because I feel like it's pointless and a slow way to do the problem. That being said... can't get the answer

Evaluate $f(4)$:
$$f(4) = \frac{4+6}{4+4}=\frac{10}{8}$$
Plug in what you know into the difference quotient:
$$\frac{\frac{x+6}{x+4}-\frac{10}{8}}{x-4}$$
Simplifying the numerator:
$$\frac{x+6}{x+4}-\frac{10}{8}=\frac{8(x+6)-10(x+4)}{8(x+4)}=\frac{8-2x}{8(x+4)}=\frac{-2(x-4)}{8(x+4)}=-\frac{(x-4)}{4(x+4)}$$
$$\frac{-\frac{x-4}{4(x+4)}}{x-4}=-\frac{x-4}{4(x+4)(x-4)}=-\frac{1}{4(x+4)}$$
