Suppose $p$ is an odd prime. Then looking mod $4$, $0\equiv (-1)^{p-1} +1$ mod $4$ but since $p$ is an odd prime , $0\equiv 2$ mod $4$, absurd So $p=2$ and so $x(x+1)(x+2)(x+3)=5040=7!$ $\implies$ $x=4$(LHS is constantly increasing so $x=4$ is the unique sol.)