decompose[m_,a_]:=With[{max = Max @ Exponent[m, a]},
    terms = SeriesCoefficient[m, {a, 0, #}]& /@ Range[0,max];
    a^Range[0,max] . MatrixForm/@terms
]

decompose[{{3+2 a+a^2,1},{2 a,a^2}}, a] //TeXForm