Solution
= Solution
The relevant <derivatives> at $x=3$ are
$$
g(3)=32,\qquad g'(3)=0,\qquad
g''(3)=0,\qquad g'''(3)=24.
$$
Thus the cubic <Taylor polynomial> is
$$
\boxed{T_3(x)=32+4(x-3)^3.}
$$
In fact the exact expansion is $g(x)=32+4(x-3)^3+(x-3)^4$, so the omitted term has order $O((x-3)^4)$ in <big O notation>.