PolynomialDivCofficientList.m

(*
PolynomialDivCofficientList

Return the coefficient list by dividing polynomials;PolynomialDivCofficientList[x^2+1,x+1,x]\[LongRightArrow]{1,-2,2},which means x^2+1==(x+1)^2-2(x+1)+2.
*)

PolynomialDivCofficientList[expr_, denom_, x_] := Module[
   {result, rem, cnt, i},
   result = {};
   rem = expr;
   cnt = Floor[Exponent[expr, x]/Exponent[denom, x]];
   For[i = cnt, i > 0, i--,
    result = result~Append~PolynomialQuotient[rem, Power[denom, i], x];
    rem = PolynomialRemainder[rem, Power[denom, i], x];
    ];
   result~Append~rem
   ];