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.