find and simplify the difference quotient $ \frac { f ( x + h ) - f ( x ) } { h } $, $ h \
eq 0 $ for the given function.
$ f ( x ) = - 2 x ^ { 2 } - x + 6 $
$ \frac { f ( x + h ) - f ( x ) } { h } = \square $ (simplify your answer.)

Question

Accepted Answer

# Explanation:
## Step1: Find $ f(x + h) $
Substitute $ x+h $ into $ f(x)=-2x^{2}-x + 6 $.
$$
\begin{align*}
f(x + h)&=-2(x + h)^{2}-(x + h)+6\
&=-2(x^{2}+2xh+h^{2})-x - h+6\
&=-2x^{2}-4xh-2h^{2}-x - h+6
\end{align*}
$$
## Step2: Calculate $ f(x + h)-f(x) $
$$
\begin{align*}
f(x + h)-f(x)&=(-2x^{2}-4xh-2h^{2}-x - h+6)-(-2x^{2}-x + 6)\
&=-2x^{2}-4xh-2h^{2}-x - h+6 + 2x^{2}+x - 6\
&=-4xh-2h^{2}-h
\end{align*}
$$
## Step3: Simplify $ \frac{f(x + h)-f(x)}{h} $
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{-4xh-2h^{2}-h}{h}\
&=\frac{h(-4x - 2h-1)}{h}\
&=-4x-2h - 1
\end{align*}
$$
# Answer:
$-4x-2h - 1$

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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: