Use the Factor Theorem to determine whether $g(x)$ is a factor of $p(x)$ in each of the following cases: $p(x) = x^3 + 3x^2 + 3x + 1$,$g(x) = x + 2$.

(D) According to the Factor Theorem,$g(x) = x - a$ is a factor of $p(x)$ if and only if $p(a) = 0$.
Here,$g(x) = x + 2$,so we set $x + 2 = 0$,which gives $x = -2$.
Now,we evaluate $p(-2)$:
$p(-2) = (-2)^3 + 3(-2)^2 + 3(-2) + 1$
$p(-2) = -8 + 3(4) - 6 + 1$
$p(-2) = -8 + 12 - 6 + 1$
$p(-2) = 4 - 6 + 1 = -1$
Since $p(-2) \neq 0$,by the Factor Theorem,$g(x)$ is not a factor of $p(x)$.