Answer
The difference quotient of the given function is equal to $-4x-2h+1$.
Work Step by Step
To find the value of $f\left( x+h \right)$,
Put $x+h$ in place of x, and then substitute in the given expression:
So,
$\begin{align}
& f\left( x+h \right)=-2{{\left( x+h \right)}^{2}}+\left( x+h \right)+10 \\
& =-2\left( {{x}^{2}}+2xh+{{h}^{2}} \right)+x+h+10 \\
& =-2{{x}^{2}}-4xh-2{{h}^{2}}+x+h+10.
\end{align}$
Now,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{-2{{x}^{2}}-4xh-2{{h}^{2}}+x+h+10-\left( -2{{x}^{2}}+x+10 \right)}{h} \\
& =\frac{-2{{x}^{2}}-4xh-2{{h}^{2}}+x+h+10+2{{x}^{2}}-x-10}{h} \\
& =\frac{-4xh-2{{h}^{2}}+h}{h} \\
& =-4x-2h+1.
\end{align}$
Therefore, the difference quotient of the given function is equal to $-4x-2h+1$.
