An integer greater than or equal to 2 that has no positive divisors other than 1 and itself is called a prime number. Answer the following questions.

\begin{itemize}
\item[(1)] Let $f(x) = x^3 + 10x^2 + 20x$. Find all integers $n$ such that $f(n)$ is a prime number.

\item[(2)] Let $a$, $b$ be integer constants, and let $g(x) = x^3 + ax^2 + bx$. Show that the number of integers $n$ such that $g(n)$ is a prime number is at most 3.
\end{itemize}