Answer
The difference quotient for the given function is $2x+h-5$
Work Step by Step
$f(x)=x^{2}-5x+8$
Find the difference quotient $\dfrac{f(x+h)-f(x)}{h}$
Start by finding $f(x+h)$. Substitute $x$ by $x+h$ in $f(x)$ and simplify:
$f(x+h)=(x+h)^{2}-5(x+h)+8=...$
$...=x^{2}+2xh+h^{2}-5x-5h+8$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{x^{2}+2xh+h^{2}-5x-5h+8-(x^{2}-5x+8)}{h}=...$
$...=\dfrac{x^{2}+2xh+h^{2}-5x-5h+8-x^{2}+5x-8}{h}=...$
$...=\dfrac{2xh+h^{2}-5h}{h}=...$
Take out common factor $h$ from the numerator and simplify:
$...=\dfrac{h(2x+h-5)}{h}=2x+h-5$
The difference quotient for the given function is $2x+h-5$
