I want check my answer
If you quote this response, you'll see how I translated the equations above into notation that's easier to read on this board. Here's your question:
Let $\displaystyle g(s) = -7s^2 + 4s + 13$.
Find $\displaystyle \frac{g(z + h) - g(z)}{h}$
This is what I found:
$\displaystyle -14z - 7h + 4$
Now, I'm not sure if that's a 'z' or a '2' in your question, so just in case, here's what I got for $\displaystyle \frac{g(2 + h) - g(2)}{h}$:
$\displaystyle -7 h - 24$
$\displaystyle g(s) = -7s^2+4s+13$
$\displaystyle g(2+h) = -7(2+h)^2 + 4(2+h) + 13 = -7(4 + 4h + h^2) + 4(2+h) + 13 = -7h^2 -24h -7$
$\displaystyle g(2) = -7(4) + 4(2) + 13 = -7$
$\displaystyle \frac{g(2+h) - g(2)}{h} = \frac{-7h^2 - 24h}{h} = \frac{h(-7h - 24)}{h} = -7h - 24$