Factorial equaling a polynomial

Question

Are there any positive integer solutions (n,x) to the equation (x)(x+1)=n! except (2,1) and (3,2)?

If not (as I suspect is the case), how do you prove that?

In general, is there a way to approach the Diophantine equation n!=P(x) where P is a polynomial?

Answer 1

This question was discussed on mathoverflow, and the question is open.