Let us begin by simplifying the expression:

(n+3)² - n²  = (n+3)(n+3) - n²

= n² + 6n + 9 - n² (expanded brackets)

= 6n + 9 (collected like terms)

= 3(2n+3) (taken out a factor of 3)

Now we can consider this simpler equivalent expression.

3 is an odd number

2n is even thus 2n+3 is odd (even plus odd is always odd)

so we have an odd*odd which is always odd, thus never even and we are done.