find and simplify the difference quotient $\frac{f(x + h)-f(x)}{h}$, $h
eq0$ for the given function. $f(x)=-x^{2}+2x + 9$ $\frac{f(x + h)-f(x)}{h}=square$

Question

Accepted Answer

# 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:
$-2x - h + 2$

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QUESTION IMAGE

Question

Explanation:

Step1: Find f(x + h)

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

Step3: Calculate the difference - quotient

Answer: