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