It's a pity to use induction

n²+n+2 = (n+1)² - n + 1

n odd => (n+1)² even => (n+1)² - n + 1 even

n even => (n+1)² odd => (n+1)² - n + 1 even

By induction

Easy for n=0

If right for n then n²+n+2 even

(n+1)²+(n+1)+2 = n²+2n+1 + n + 3 = n²+n+2 +2n+2= n²+n+2 +2(n+1)