Answer
The difference quotient for the provided function is \[6x+3h+1\].
Work Step by Step
Consider the provided function: $f\left( x \right)=3{{x}^{2}}+x+5$.
Now, substitute $x=x+h$ in the above equation to find $f\left( x+h \right)$
That is,
$\begin{align}
& f\left( x+h \right)=3{{\left( x+h \right)}^{2}}+\left( x+h \right)+5 \\
& =3{{x}^{2}}+6xh+3{{h}^{2}}+x+h+5
\end{align}$
Now, apply the difference quotient formula,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{3{{x}^{2}}+6xh+3{{h}^{2}}+x+h+5-\left( 3{{x}^{2}}+x+5 \right)}{h} \\
& =\frac{3{{x}^{2}}+6xh+3{{h}^{2}}+x+h+5-3{{x}^{2}}-x-5}{h} \\
& =\frac{6xh+3{{h}^{2}}+h}{h} \\
& =\frac{h\left( 6x+3h+1 \right)}{h}
\end{align}$
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 $6x+3h+1$.
