OrePolynomial[{{t^2 - 4*t^3 - 16*t^4 + 64*t^5, {3}}, 
  {5*t - 20*t^2 - 112*t^3 + 448*t^4, {2}}, {4*(1 - 5*t - 44*t^2 + 160*t^3), 
   {1}}, {4*(-1 - 12*t + 32*t^2), {0}}}, OreAlgebraObject[{Der[t]}, Together, 
  Together[#1 + #2] & , Together[#1*#2] & , None], DegreeLexicographic]
