Expressing elements of a matrix over a polynomial ring as polynomials

Question

Let $S$ be a ring and let $\theta(x) \in M_n(S[x])$. Then we can write $\theta(x)= \Sigma_0^p \alpha_i x^i$ where $\alpha_i \in M_n(S)$.

This is happening in Lemma 16.4 in Donald Passmans "A Course in Ring Theory." Could somebody help me to see how we can write the matrix like this? Thanks

Answer 1

E.g. $$ \pmatrix{x^2+2x+1&1\\ -x^3-x-1&-4x^2+3x} =\pmatrix{0&0\\ -1&0}x^3 +\pmatrix{1&0\\ 0&-4}x^2 +\pmatrix{2&0\\ -1&0}x +\pmatrix{1&1\\ -1&0}. $$