Answer
The difference quotient for the provided function is \[-4x-2h-1\].
Work Step by Step
Consider the provided function: $f\left( x \right)=-2{{x}^{2}}-x+3$.
Now, substitute $x=x+h$ in the above equation to find $f\left( x+h \right)$
That is,
$\begin{align}
& f\left( x+h \right)=-2{{\left( x+h \right)}^{2}}-\left( x+h \right)+3 \\
& =-2{{x}^{2}}-4xh-2{{h}^{2}}-x-h+3
\end{align}$
Now, apply the difference quotient formula,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{-2{{x}^{2}}-4xh-2{{h}^{2}}-x-h+3-\left( -2{{x}^{2}}-x+3 \right)}{h} \\
& =\frac{-2{{x}^{2}}-4xh-2{{h}^{2}}-x-h+3+2{{x}^{2}}+x-3}{h} \\
& =\frac{-4xh-2{{h}^{2}}-h}{h} \\
& =\frac{h\left( -4x-2h-1 \right)}{h}
\end{align}$
Further solve and get,
$\frac{f\left( x+h \right)-f\left( x \right)}{h}=-4x-2h-1$
Hence, the difference quotient for the provided function is $-4x-2h-1$.
