Answer
The difference quotient for the given function is $-6x-3h+2$
Work Step by Step
$f(x)=-3x^{2}+2x-1$
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:
$f(x+h)=-3(x+h)^{2}+2(x+h)-1=...$
$...=-3(x^{2}+2xh+h^{2})+2x+2h-1=...$
$...=-3x^{2}-6xh-3h^{2}+2x+2h-1$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{-3x^{2}-6xh-3h^{2}+2x+2h-1-(-3x^{2}+2x-1)}{h}=...$
$...=\dfrac{-3x^{2}-6xh-3h^{2}+2x+2h-1+3x^{2}-2x+1}{h}=...$
$...=\dfrac{-6xh-3h^{2}+2h}{h}=...$
Take out common factor $h$ from the numerator and simplify:
$...=\dfrac{h(-6x-3h+2)}{h}=-6x-3h+2$
The difference quotient for the given function is $-6x-3h+2$
