find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$, for the function below.\n\n$f(x)=-x^{…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=-(x + h)^2-2(x + h)+5\ &=-(x^{2}+2xh+h^{2})-2x-2h + 5\ &=-x^{2}-2xh - h^{2}-2x-2h + 5 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}-2x-2h + 5)-(-x^{2}-2x + 5)\ &=-x^{2}-2xh - h^{2}-2x-2h + 5 + x^{2}+2x - 5\ &=-2xh - h^{2}-2h \end{align*} ]

Step3: Find the difference quotient $\frac{f(x + h)-f(x)}{h}$

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

Answer:

$-2x - h - 2$