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