Note that $n(n+1)\cdot (n+1)(n+2)=n(n+2)\cdot (n+1)^2=[(n+1)^2-1]\cdot (n+1)^2$,so $n(n+1)=\frac{[(n+1)^2-1]\cdot (n+1)^2}{(n+1)(n+2)}$.Thus every almost square can be expressed as a quotients as two almost squares.

silouan wrote: We say that a positive integer is an almost square, if it is equal to the product of two consecutive positive integers. Prove that every almost square can be expressed as a quotient of two almost squares. V. Senderov What an easy problem for ARMO $a(a+1)$=$\frac{(a-1)(a+1)a^2}{(a-1)a}$