OrePolynomial[{{t^2 - 5*t^3 - 30*t^4 + 40*t^5 + 64*t^6, {3}}, 
  {5*t - 33*t^2 - 252*t^3 + 200*t^4 + 576*t^5, {2}}, 
  {4*(1 - 12*t - 117*t^2 + 22*t^3 + 288*t^4), {1}}, 
  {12*(-1 - 12*t - 6*t^2 + 32*t^3), {0}}}, OreAlgebraObject[{Der[t]}, 
  Together, Together[#1 + #2] & , Together[#1*#2] & , None], 
 DegreeLexicographic]
