OrePolynomial[{{-4*(256*t^2 - 870*t^3 + 1055*t^4 - 540*t^5 + 99*t^6), {3}}, 
  {-4*(-256*t + 588*t^2 - 210*t^3 - 189*t^4 + 99*t^5), {2}}, 
  {3*(1024 - 4512*t + 7293*t^2 - 4770*t^3 + 1089*t^4), {1}}, 
  {12*(-96 + 307*t - 306*t^2 + 99*t^3), {0}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
