Smith Normal Form of this polynomial non square matrix in $\mathbb Q[t]$ 
Given $$A=\begin{bmatrix}t^3-t^2&t^2-t\\2t^3&t^4-t\\t&2t^3-2t^2\end{bmatrix} \in M_{3\times 2}(\mathbb Q[t])$$
  I'm looking for the matrices $S,P,Q$ such that $S=QAP$, where $Q$ are the row and $P$ are the column transformations and $S$ is its SNF.

I have tried using the euclidean algorithm, but I can't reach SNF, since there happen to be ridiculously high fractions. After that step I struggle to find any reasonable answer:
$$ \begin{bmatrix}t&0\\0&-4t^5+5t^4-t\\0&-2t^5+4t^4-2t^3+t^2-t\end{bmatrix}$$
Any help is appreciated.

Since $-4t^5+5t^4-t=-t(t-1)(4t^3-t^2-t-1)$ and $-2t^5+4t^4-2t^3+t^2-t=-t(t-1)(2t^3-2t^2-1)$, you only need to find an integer multiple of the GCD of $4t^3-t^2-t-1$ and $2t^3-2t^2-1$ using Euclidean algorithm. This should be straightforward:
$$
\begin{align}
4t^3-t^2-t-1 &= 2(2t^3-2t^2-1) + (3t^2-t+1)\\
9(2t^3-2t^2-1) &= (6t-4)(3t^2-t+1) - 5(2t+1)\\
4(3t^2-t+1) &= (6t-5)(2t+1) + 9.
\end{align}
$$
