By Definition MM,
\begin{align*}
AB&=
\left[
\left.
\begin{bmatrix}
2&5\\
-1&3\\
2&-2
\end{bmatrix}
\colvector{1\\2}
\right\vert\
\left.
\begin{bmatrix}
2&5\\
-1&3\\
2&-2
\end{bmatrix}
\colvector{5\\0}
\right\vert\
\left.
\begin{bmatrix}
2&5\\
-1&3\\
2&-2
\end{bmatrix}
\colvector{-3\\2}
\right\vert\
\left.
\begin{bmatrix}
2&5\\
-1&3\\
2&-2
\end{bmatrix}
\colvector{4\\-3}
\right.
\right]
\end{align*}
Repeated applications of Definition MVP give
\begin{align*}
&=
\left[
\left.1\colvector{2\\-1\\2}+2\colvector{5\\3\\-2}\right\vert\
\left.5\colvector{2\\-1\\2}+0\colvector{5\\3\\-2}\right\vert\
\left.-3\colvector{2\\-1\\2}+2\colvector{5\\3\\-2}\right\vert\
\left.4\colvector{2\\-1\\2}+(-3)\colvector{5\\3\\-2}\right.
\right]\\
&=
\begin{bmatrix}
12 & 10 & 4 & -7\\
5 & -5 & 9 & -13\\
-2 & 10 & -10 & 14
\end{bmatrix}
\end{align*}