Would the Smith Normal Form of the following matrix over $\mathbb Q[x]$
\begin{pmatrix} (x+a)(x+b) & 0 & 0 &0 \\ 0 & (x+c)(x+d) & 0 & 0 \\ 0 &0 & x^3(x+a) & 0 \\ 0 & 0 & 0& x^2(x+b) \end{pmatrix}
simply be
\begin{pmatrix} f(x) & 0 & 0 &0 \\ 0 & f(x) & 0 & 0 \\ 0 &0 & f(x) & 0 \\ 0 & 0 & 0& f(x) \end{pmatrix}
where $f(x)= x^3(x+a)(x+b)(x+c)(x+d)$?
I am not sure because that would make the question quite trivial.