#### Answer

The difference quotient for the given function is $-2x-h+2$

#### Work Step by Step

$f(x)=-x^{2}+2x+4$
Find the difference quotient $\dfrac{f(x+h)-f(x)}{h}$
Start by finding $f(x+h)$. Substitute $x$ by $x+h$ in $f(x)$ and simplify:
$f(x+h)=-(x+h)^{2}+2(x+h)+4=...$
$...=-(x^{2}+2xh+h^{2})+2x+2h+4=...$
$...=-x^{2}-2xh-h^{2}+2x+2h+4$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{-x^{2}-2xh-h^{2}+2x+2h+4-(-x^{2}+2x+4)}{h}=...$
$...=\dfrac{-x^{2}-2xh-h^{2}+2x+2h+4+x^{2}-2x-4}{h}=...$
$...=\dfrac{-2xh-h^{2}+2h}{h}=...$
Take out common factor $h$ from the numerator and simplify:
$...=\dfrac{h(-2x-h+2)}{h}=-2x-h+2$
The difference quotient for the given function is $-2x-h+2$