For all integers n; n^2+n+1 is odd. Prove the statement directly from definition of the terms & do not use any previously known facts. My main problem is if n^2+n+1=n(n+1)+1 then can I say n is even & n+1 is odd or would that imply that I'm using previously known facts. In the text there is a section on parity, but I'd hate to have to have a proof inside of a proof. Then even worse, would I then have to go through & prove what even & odd are? Not really sure what "directly from the definition of the terms" implies. Any ideas? Its obvious the professor would know best what she meant but there is no way to contact her atm.