find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, h $\neq$ 0, for the following…

Explanation:

Step1: Find f(x + h)

Substitute x+h into f(x): [ \begin{align*} f(x + h)&=-2(x + h)^2+6(x + h)+2\ &=-2(x^{2}+2xh+h^{2})+6x + 6h+2\ &=-2x^{2}-4xh-2h^{2}+6x + 6h+2 \end{align*} ]

Step2: Calculate f(x + h)-f(x)

[ \begin{align*} f(x + h)-f(x)&=(-2x^{2}-4xh-2h^{2}+6x + 6h+2)-(-2x^{2}+6x + 2)\ &=-2x^{2}-4xh-2h^{2}+6x + 6h+2 + 2x^{2}-6x - 2\ &=-4xh-2h^{2}+6h \end{align*} ]

Step3: Find the difference quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-4xh-2h^{2}+6h}{h}\ &=\frac{h(-4x-2h + 6)}{h}\ &=-4x-2h + 6 \end{align*} ]

Answer:

$-4x-2h + 6$