How to proof for primes

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.