TITLE: Why does $p^2+8$ prime imply $p^3+4$ prime
QUESTION [12 upvotes]: How do I prove that $p^2+8$ prime implies that $p^3+4$ is prime? What is the general pattern of thought for these problems?

REPLY [2 votes]: The question or (at least the current revision) does not say that $p$ is prime. Although this is very probably the intended meaning. (The letter $p$ is typically used to denote prime numbers. Exactly this exercise with the assumption the $p$ is a prime number appears in many introductory texts in number theory.)
If we take the question in the way it is stated, i.e., we are asking whether $p^2+8$ being prime necessarily implies $p^3+4$ being prime, then the claim is not true.
The arguments in the other answers show us that if want to find a counterexample, it is possible only with $p$ being odd multiple of $3$. And indeed we can find a counterexample:
$$\begin{align}
15^2+8&=233\\
15^3+4&=3379
\end{align}
$$
233 is a prime number, but 3379 is not a prime, in fact we get the factorization $3379=31\cdot109$.