OrePolynomial[{{-t^3 + 2*t^5 - 2*t^9 + t^11, {3}}, 
  {4*(t^2 - 2*t^4 + 2*t^6 - 2*t^8 + t^10), {2}}, 
  {-5*t + 14*t^3 - 8*t^5 + 2*t^7 - 3*t^9, {1}}, 
  {5 - 30*t^2 + 56*t^4 - 18*t^6 + 3*t^8, {0}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
