Answer
The difference quotient for the given function is $-4x-2h+5$
Work Step by Step
$f(x)=-2x^{2}+5x+7$
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:
$f(x+h)=-2(x+h)^{2}+5(x+h)+7=...$
$...=-2(x^{2}+2xh+h^{2})+5x+5h+7=...$
$...=-2x^{2}-4xh-2h^{2}+5x+5h+7$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{-2x^{2}-4xh-2h^{2}+5x+5h+7-(-2x^{2}+5x+7)}{h}=...$
$...=\dfrac{-2x^{2}-4xh-2h^{2}+5x+5h+7+2x^{2}-5x-7}{h}=...$
$...=\dfrac{-4xh-2h^{2}+5h}{h}=...$
Take out common factor $h$ from the numerator and simplify:
$...=\dfrac{h(-4x-2h+5)}{h}=-4x-2h+5$
The difference quotient for the given function is $-4x-2h+5$
