Proof that for any integer $n>1$ the number $n^5+n^4+1$ is not a prime.
Simply rewrite $n^5+n^4+1$ as $n^5+n^4+n^3-n^3-n^2-n+n^2+n+1$
Factor out common terms
$n^3(n^2+n+1)-n(n^2+n+1)+(n^2+n+1)$
$(n^3-n+1)(n^2+n+1)$
The product of two integers greater than $1$, hence $n^5+n^4+1$ is not a prime.
Reference
Andreescu, T., & Andrica, D. "Number Theory Structures, Examples, and Problems" page 25.
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