為什麼四個連續自然數的積再加１，一定是個完全平方數？

Let the smallest one be x

x(x+1)(x+2)(x+3) + 1
= (x^2+x)(x^2+5x+6)+1
= x^4+6x^3+11x^2+6x+1
= (x^2+3x+1)^2

which is a complete square

n(n+1)(n+2)(n+3) +1
=n(n^2 +3n +2)(n+3) +1
=n(n^3 +6n^2 +11n +6) +1
=n^4 +6n^3 +11n^2 +6n +1
=(n^2 +3n +1)^2