find and simplify the difference quotient $ \frac { f ( x + h ) - f ( x ) } { h } $, $ h \
eq 0 $ for the given function.
$ f ( x ) = - x ^ { 2 } + 4 x + 5 $
$ \frac { f ( x + h ) - f ( x ) } { h } = \square $

Question

Accepted Answer

# Explanation:
## Step1: Find $ f(x + h) $
Substitute $ x+h $ into $ f(x)=-x^{2}+4x + 5 $.
$$
\begin{align*}
f(x + h)&=-(x + h)^{2}+4(x + h)+5\
&=-(x^{2}+2xh+h^{2})+4x + 4h+5\
&=-x^{2}-2xh - h^{2}+4x + 4h+5
\end{align*}
$$
## Step2: Calculate $ f(x + h)-f(x) $
$$
\begin{align*}
f(x + h)-f(x)&=(-x^{2}-2xh - h^{2}+4x + 4h+5)-(-x^{2}+4x + 5)\
&=-x^{2}-2xh - h^{2}+4x + 4h+5 + x^{2}-4x - 5\
&=-2xh - h^{2}+4h
\end{align*}
$$
## Step3: Simplify $ \frac{f(x + h)-f(x)}{h} $
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{-2xh - h^{2}+4h}{h}\
&=\frac{h(-2x - h + 4)}{h}\
&=-2x - h+4
\end{align*}
$$

# Answer:
$-2x - h + 4$

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QUESTION IMAGE

Question

Explanation:

Step1: Find \( f(x + h) \)

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

Step3: Simplify \( \frac{f(x + h)-f(x)}{h} \)

Answer: