Answer
The difference quotient for the provided function is \[-2x-h+2\].
Work Step by Step
Consider the provided function: $f\left( x \right)=-{{x}^{2}}+2x+4$.
Now, substitute $x=x+h$ in the above equation to find $f\left( x+h \right)$
That is,
$\begin{align}
& f\left( x+h \right)=-{{\left( x+h \right)}^{2}}+2\left( x+h \right)+4 \\
& =-{{x}^{2}}-2xh-{{h}^{2}}+2x+2h+4
\end{align}$
Now, apply the difference quotient formula,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{-{{x}^{2}}-2xh-{{h}^{2}}+2x+2h+4-\left( -{{x}^{2}}+2x+4 \right)}{h} \\
& =\frac{-{{x}^{2}}-2xh-{{h}^{2}}+2x+2h+4+{{x}^{2}}-2x-4}{h} \\
& =\frac{-2xh-{{h}^{2}}+2h}{h} \\
& =\frac{h\left( -2x-h+2 \right)}{h}
\end{align}$
Further solve and get,
$\frac{f\left( x+h \right)-f\left( x \right)}{h}=-2x-h+2$
Hence, the difference quotient for the provided function is $-2x-h+2$.
