Prove that if $p$ and $p^2+2$ are prime then $p^3+2$ is prime too

Question

I'm trying to figure out how to prove that if $p$ and $p^2+2$ are prime numbers then $p^3+2$ is a prime number too. Can someone help me please?

Answer 1

If $p=2$, then $p^2+2$ is not prime.

If $p=3$, then $p^2+2 = 11$, then $p^3+2=29$ is prime.

If $p>3$, then $p \equiv \pm 1 \pmod 3$, then $p^2+2 \equiv 0 \pmod 3$. So, $p^2+2$ is not prime.