jmc

We successively expand by multiplying binomials: $$\begin{array}{rl}(a& -1)(a+1)(a+2)-(a-2)(a+1)\\ & =(a^2-1)(a+2)-(a-2)(a+1)\\ & =(a^3+2a^2-a-2)-(a^2-a-2)\\ & =a^3+a^2.\end{array}$$So our answer is just $\boxed{a^3+a^2}$.