Answer
The difference quotient for the given function is $2x+h-4$
Work Step by Step
$f(x)=x^{2}-4x+3$
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:
$f(x+h)=(x+h)^{2}-4(x+h)+3=...$
$...=x^{2}+2xh+h^{2}-4x-4h+3$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{x^{2}+2xh+h^{2}-4x-4h+3-(x^{2}-4x+3)}{h}=...$
$...=\dfrac{x^{2}+2xh+h^{2}-4x-4h+3-x^{2}+4x-3}{h}=...$
$...=\dfrac{2xh+h^{2}-4h}{h}=...$
Take out common factor $h$ from the numerator and simplify:
$...=\dfrac{h(2x+h-4)}{h}=2x+h-4$
The difference quotient for the given function is $2x+h-4$