Answer
The difference quotient for the provided function is $2x+h-5$.
Work Step by Step
Consider the provided function: $f\left( x \right)={{x}^{2}}-5x+8$.
Now, substitute $x=x+h$ in the above equation to find $f\left( x+h \right)$
That is,
$\begin{align}
& f\left( x+h \right)={{\left( x+h \right)}^{2}}-5\left( x+h \right)+8 \\
& ={{x}^{2}}+2xh+{{h}^{2}}-5x-5h+8
\end{align}$
Now, apply the difference quotient formula,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{{{x}^{2}}+2xh+{{h}^{2}}-5x-5h+8-\left( {{x}^{2}}-5x+8 \right)}{h} \\
& =\frac{{{x}^{2}}+2xh+{{h}^{2}}-5x-5h+8-{{x}^{2}}+5x-8}{h} \\
& =\frac{h\left( 2x+h-5 \right)}{h} \\
& =2x+h-5
\end{align}$
Hence, the difference quotient for the provided function is $2x+h-5$.
