Simplify $(a-1)(a+1)(a+2) - (a-2)(a+1).$
We successively expand by multiplying binomials: \begin{align*}
(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{align*}So our answer is just $\boxed{a^3 + a^2}$.