smc

Factoring the polynomial gives $(n+1)(n)(n-1)$ According to the factorization, one of those factors must be a multiple of two because there are more than 2 consecutive integers. In addition, because there are three consecutive integers, one of the integers must be a multiple of 3. Therefore $6$ must divide the given expression. Plugging in $n=2$ yields $6$. So the largest possibility is $6$. Clearly the answer is $\boxed{6}$