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Notice that $n^5 + n^4 + 1 = (n^2 + n + 1)(n^3 - n + 1)$.

That means, for all $n^2 + n + 1 \neq 1$ and $n^3 - n + 1 \neq 1$, $n^5 + n^4 + 1$ is composite.

This means when $n^5 + n^4 + 1$ is prime, $n^2 + n = 0$ or $n^3 = n$

Solving, we get n = -1 (rejected) or n = 0 (rejected) or n = 1.

When n = 1, $n^5 + n^4 + 1 = 3$, which is a prime.

The only possibility is n = 1.