Answer
The difference quotient for the given function is $4x+2h+1$
Work Step by Step
$f(x)=2x^{2}+x-1$
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:
$f(x+h)=2(x+h)^{2}+x+h-1=...$
$...=2(x^{2}+2xh+h^{2})+x+h-1=...$
$...=2x^{2}+4xh+2h^{2}+x+h-1$
Substitute the known values into the formula for the difference quotient:
$\dfrac{f(x+h)-f(x)}{h}=...$
$...=\dfrac{2x^{2}+4xh+2h^{2}+x+h-1-(2x^{2}+x-1)}{h}=...$
$...=\dfrac{2x^{2}+4xh+2h^{2}+x+h-1-2x^{2}-x+1}{h}=...$
$...=\dfrac{4xh+2h^{2}+h}{h}$
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 $4x+2h+1$
