Answer
$$
f(x) =-4 x^{2}+3 x+2
$$
(a)
$$
\begin{aligned} f(x+h) =-4 x^{2}-8 h x-4 h^{2}+3 x+3 h+2
\end{aligned}
$$
(b)
$$
\begin{split}
f(x+h)-f(x) =-8 h x-4 h^{2}+3 h
\end{split}
$$
(c)
$$
\begin{split}
\frac{f(x+h)-f(x)}{h} =-8 x-4 h+3
\end{split}
$$
Work Step by Step
$$
f(x) =-4 x^{2}+3 x+2
$$
(a)
$$
\begin{aligned} f(x+h) &=-4(x+h)^{2}+3(x+h)+2 \\
&=-4\left(x^{2}+2 h x+h^{2}\right)+3 x+3 h+2 \\
&=-4 x^{2}-8 h x-4 h^{2}+3 x+3 h+2
\end{aligned}
$$
(b)
$$
\begin{split}
f(x+h)-f(x) & = -4 x^{2}-8 h x-4 h^{2}+3 x+3 h+2 \\
& \qquad-\left(-4 x^{2}+3 x+2\right) \\
& =-4 x^{2}-8 h x-4 h^{2}+3 x+3 h+2 \\
& \qquad+4 x^{2}-3 x-2 \\
&=-8 h x-4 h^{2}+3 h
\end{split}
$$
(c)
$$
\begin{split}
\frac{f(x+h)-f(x)}{h} & = \frac{-8 h x-4 h^{2}+3 h}{h} \\
&=\frac{h(-8 x-4 h+3)}{h} \\
&=-8 x-4 h+3
\end{split}
$$
