I would like to put this matrix below into Smith Normal Form over $\mathbb{Q}[x]: $ $$\left( \begin{array}{ccc} 7 & x & 0 & -x \\ 0 & x-3 & 0 & 3\\ 0 & 0 & x-4 & 0 \\ x-6 & -1 & 0 & x+1 \end{array} \right)$$
but I am stuck here:
$$\left( \begin{array}{ccc} 7 & 0 & 0 & -x \\ 0 & x & 0 & 3\\ 0 & 0 & x-4 & 0 \\ 8x+22 & x & 0 & -x^2-3x+1 \end{array} \right)$$
I'm not sure how to proceed. Any help is appreciated.
Here are the first few steps as per my suggestion in the comments: \begin{align*} \left(\begin{array}{rrrr} x & 7 & 0 & -x \\ x - 3 & 0 & 0 & 3 \\ 0 & 0 & x - 4 & 0 \\ -1 & x - 6 & 0 & x + 2 \end{array}\right) &\leadsto \left(\begin{array}{rrrr} -1 & x - 6 & 0 & x + 2 \\ x - 3 & 0 & 0 & 3 \\ 0 & 0 & x - 4 & 0 \\ x & 7 & 0 & -x \end{array}\right)\\ \left(\begin{array}{rrrr} 1 & -x + 6 & 0 & -x - 2 \\ x - 3 & 0 & 0 & 3 \\ 0 & 0 & x - 4 & 0 \\ x & 7 & 0 & -x \end{array}\right) &\leadsto \left(\begin{array}{rrrr} 1 & -x + 6 & 0 & -x - 2 \\ 0 & x^{2} - 9 x + 18 & 0 & x^{2} - x - 3 \\ 0 & 0 & x - 4 & 0 \\ 0 & x^{2} - 6 x + 7 & 0 & x^{2} + x \end{array}\right)\\ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & x^{2} - 9 x + 18 & 0 & x^{2} - x - 3 \\ 0 & 0 & x - 4 & 0 \\ 0 & x^{2} - 6 x + 7 & 0 & x^{2} + x \end{array}\right) &\leadsto \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & x - 4 & 0 & 0 \\ 0 & 0 & x^{2} - 9 x + 18 & x^{2} - x - 3 \\ 0 & 0 & x^{2} - 6 x + 7 & x^{2} + x \end{array}\right) \end{align*} Can you take it from here? In the end, I get $$ \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & x^{4} - 3 x^{3} - 11 x^{2} + 7 x + 84 \end{array}\right) $$ for the Smith normal form.