"simplify the difference quotient f(x+h)-f(x) over h whenever f(x)=2x^2+4x+2"
First expand $\displaystyle (x+h)^2$
$\displaystyle (x+h)^2=x^2+2hx+h^2$
Thus, the difference quotient becomes $\displaystyle \frac{2(x^2+2hx+h^2)+4(x+h)+2-2x^2-4x-2}{h}$ $\displaystyle =\frac{2x^2+4hx+2h^2+4x+4h+2-2x^2-4x-2}{h}$
Try to take it from here. See if you can finish simplifying this expression.
The remaining terms in the numerator should have at least one h.
--Chris