Algebraic Proofs - Help with reasoning

I'm stuck on my reasoning for this algebraic proof:
Prove algebraically that the sum of the squares of any two consecutive numbers always leaves a remainder of 1 when divided by 4.

My working:

Two consecutive numbers = n and n+1
(n)^2 + (n+1)^2 = n^2+n^2+n+n+1
= 2n^2+2n+1
I've then factorised the 2n out to get:
2n(n+1)+1

How do I then explain my reasoning of why there is a remainder of 1 when the term is divided by 4?

If you can show that n(n+1) is even then that must mean that 2n(n+1) is a multiple of 4. Think about why n(n+1) must be even.

Because either n or n+1 must be even, so the other must be odd. Even number × odd number = even number. Is that it?

Yes that's right.