Answer
The difference quotient for the provided function is \[-6x-3h+1\].
Work Step by Step
Consider the provided function: $f\left( x \right)=-3{{x}^{2}}+x-1$.
Now, substitute $x=x+h$ in the above equation to find $f\left( x+h \right)$
That is,
$\begin{align}
& f\left( x+h \right)=-3{{\left( x+h \right)}^{2}}+\left( x+h \right)-1 \\
& =-3{{x}^{2}}-6xh-3{{h}^{2}}+x+h-1
\end{align}$
Now, apply the difference quotient formula,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{-3{{x}^{2}}-6xh-3{{h}^{2}}+x+h-1-\left( -3{{x}^{2}}+x-1 \right)}{h} \\
& =\frac{-2{{x}^{2}}-4xh-2{{h}^{2}}-x-h+3+2{{x}^{2}}+x-3}{h} \\
& =\frac{-6xh-3{{h}^{2}}+h}{h} \\
& =\frac{h\left( -6x-3h+1 \right)}{h}
\end{align}$
Further solve and get,
$\frac{f\left( x+h \right)-f\left( x \right)}{h}=-6x-3h+1$
Hence, the difference quotient for the provided function is $-6x-3h+1$.
