You presumably mean (n^2)-1 =(n+1)(n-1).
If n is odd then n+1 and n-1 are consecutive even numbers
so the expression can be re-written 2a (2a +2) or 4
(a sqd +a) where a is any positive integer.
But (a sqd +a)=a(a+1) and either a or a+1 must be even giving another
Hence the expression is divisible by 4