Help - Prove using cases

Question

for every natural number n ≥ 1 the number $n^2 + n + 4$ is not a prime number

Hi there, I am trying to prove this using a proof by cases. I am just simply confused on how to do this or start this off.

Any help is appreciated.

Answer 1

$$n^2+n+4=n(n+1)+4$$ $n(n+1)$ is always even, due to which the above expression is also even .

Also, $$n^2+n+4\ge 1+1+4 = 6$$ Thus it can never be a prime number since the only even prime number is $2$