OrePolynomial[{{-2*t^4 - t^6 + 6*t^8 - 4*t^12 + t^14, {4}}, 
  {4*(-t^3 + 4*t^5 + 11*t^7 - 2*t^9 - 16*t^11 + 4*t^13), {3}}, 
  {3*(2*t^2 + 13*t^4 + 34*t^6 - 44*t^8 - 92*t^10 + 23*t^12), {2}}, 
  {3*(-2*t - 7*t^3 + 30*t^5 - 100*t^7 - 108*t^9 + 27*t^11), {1}}, 
  {3*(2 + 7*t^2 + 2*t^4 - 28*t^6 - 20*t^8 + 5*t^10), {0}}}, 
 OreAlgebraObject[{Der[t]}, Together, Together[#1 + #2] & , 
  Together[#1*#2] & , None], DegreeLexicographic]
