Answer
The difference quotient for the given function is $-4x-2h-1$
Work Step by Step
$f(x)=-2x^{2}-x+3$
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:
$f(x+h)=-2(x+h)^{2}-(x+h)+3=...$
$...=-2(x^{2}+2xh+h^{2})-x-h+3=...$
$...=-2x^{2}-4xh-2h^{2}-x-h+3$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{-2x^{2}-4xh-2h^{2}-x-h+3-(-2x^{2}-x+3)}{h}=...$
$...=\dfrac{-2x^{2}-4xh-2h^{2}-x-h+3+2x^{2}+x-3}{h}=...$
$...=\dfrac{-4xh-2h^{2}-h}{h}=...$
Take out common factor $h$ from the numerator and simplify:"
$...=\dfrac{h(-4x-2h-1)}{h}=-4x-2h-1$
The difference quotient for the given function is $-4x-2h-1$
