find and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$, $h\neq0$ for the given function…

Explanation:

Step1: Find f(x + h)

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

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

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

Step3: Calculate the difference quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{-4xh-2h^{2}-h}{h}\ &=\frac{h(-4x-2h - 1)}{h}\ &=-4x-2h - 1 \end{align*} ]

Answer:

$-4x-2h - 1$