The matrix $$A=\begin{pmatrix}1&1&2&0\\-2&1&-1&-3\\3&0&3&3\\-4&-2&-6&-6\\-1&0&-1&0\end{pmatrix}$$ transforms via Gaussian elimination to $$\begin{pmatrix}1&0&1&0\\0&1&1&0\\0&0&0&1\\0&0&0&0\\0&0&0&0\end{pmatrix}.$$
(This is given in the exercise.) I have calculated a basis of the $1$-dimensional kernel of $A$, that is $v=(1,1,-1,0).$
Thus I get a basis $B=(e_1,e_2,e_4,v)$ of $\mathbb{R}^4$. Here $e_i$ is the $i$-th standard basis vector.
I also get a basis $B'=(Ae_1,Ae_2,Ae_4,e_2,e_5)$ of $\mathbb{R}^5$.
Does this help to find invertible matrices $S,T$ such that $T^{-1}AS$ has Smith normal form?