If n is a natural number , then n(n +1) (n +2) (n +3) +1 is always: 

The expression $n(n+1)(n+2)(n+3)+1$ can be rewritten as
$ n(n+3)(n+1)(n+2)+1 = (n^2+3n)(n^2+3n+2)+1 $.

Let $x = n^2 + 3n$.
Then the expression becomes
$ x(x+2)+1 = x^2 + 2x + 1 = (x+1)^2 $.

So, $n(n+1)(n+2)(n+3)+1$ is always a perfect square.