How to factorize $n(n+1)(n+2)(n+3)+1$? How to factorize $n(n+1)(n+2)(n+3)+1$ ? It's turned into the lowest power of $n$ ever possible but the question wants me to factorize it. How can I do that ?

$$n(n+3)=n^2+3n$$

and $$(n+1)(n+2)=n^2+3n+2$$

$$\implies n(n+1)(n+2)(n+3)=(n^2+3n)[(n^2+3n)+2]+1=[(n^2+3n)+1]^2$$