OrePolynomial[{{-t^2 + 2*t^3 + 3*t^4, {3}}, {3*(-2*t + 5*t^2 + 9*t^3), {2}}, 
  {6*(-1 + 4*t + 9*t^2), {1}}, {6*(1 + 3*t), {0}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
