# For every positive integer $n, n^2 + 4n + 3$ is not a prime

Prove: For every positive integer $n, n^2 + 4n + 3$ is not a prime.

I tried to disprove the statement, which I could not using several number examples with constructive proof.

However I am not sure how to correctly step by step prove it.

Hint: $$n^2+ 4n+3=(n+1)(n+3)$$

• YEs got it! thats basically what I needed! Dec 22, 2014 at 16:11

Hint: $n^2+4n+3 = (n+1)(n+3)$. Can you take it further?

• yes, thank you! Dec 22, 2014 at 16:12

it is $$(n+1)(n+3)$$ and thus not prime because it has to factors greater than one.

Recall that $$n^2 + 4n + 3 = (n + 1)(n + 3)$$. Since $$n$$ is necessarily a positive integer, it is obvious that $$n^2 + 4n + 3$$ is composite.