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)&=-(x + h)^2+2(x + h)+9\ &=-(x^{2}+2xh+h^{2})+2x + 2h+9\ &=-x^{2}-2xh - h^{2}+2x + 2h+9 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}+2x + 2h+9)-(-x^{2}+2x + 9)\ &=-x^{2}-2xh - h^{2}+2x + 2h+9 + x^{2}-2x - 9\ &=-2xh - h^{2}+2h \end{align*} ]

Step3: Calculate the difference - quotient

[ \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)