If $P(n)$ is the above polynomial and $\omega$ is a primitive fifth root of unity, it is easy to see that $\{\omega^{5n-1},\omega^{5n-2},\omega^{5n-3},\omega,1\}=\{1,\omega,...\omega^4\}$. Therefore, $P(\omega)=0$ for all primitive roots of unity. This implies that the polynomial $n^4+n^4+n^2+n+1$ (which the product of factors of the form $(n-\omega)$). Therefore $n^4+n^3+n^2+n+1|P(n)$. If $n\geq 2$ the right-hand side is actually greater (because $n^{5n-1}>n^4$ for $n\geq 2$). Therefore $P(n)$ can't be prime.