a=(x^2)+2x;
b=x-c;

Resultant[a,b,x]//Expand

Resultantenfunktion

Clear[resultant]
resultant[a_,b_,x_] := (b^(Exponent[a,x]))/;FreeQ[b,x]

resultant[a_,b_,x_] :=(((-1))^((Exponent[a,x]*Exponent[b,x])))*resultant[b,a,x]/; Exponent[a,x]<Exponent[b,x]

resultant[a_,b_,x_]:=Module[{m,n,r},
n=Exponent[a,x];
m=Exponent[b,x];
r=PolynomialRemainder[a,b,x];
(((-1))^((n*m)))*((Coefficient[b,x,m])^((n-Exponent[r,x])))*resultant[b,r,x]
]

resultant[a,b,x]

Sylvestermatrix

Clear[sylvesterMatrix]
sylvesterMatrix[a_,b_,x_]:=
Module[{n,m,alist,blist,amat,bmat},
n=Exponent[a,x];
m=Exponent[b,x];
alist=PadRight[Reverse[CoefficientList[a,x]],n+m];
blist=PadRight[Reverse[CoefficientList[b,x]],n+m];
If[m===0,amat={},amat=NestList[RotateRight,alist,m-1]];
If[n===0,bmat={},bmat=NestList[RotateRight,blist,n-1]];
Join[amat,bmat]
]

Sylvestermatrix-Beispiele

d=(∑_(k=0)^7)(((k+1))^3)*(x^k)
e=(∑_(k=0)^5)k*(((-1))^k)*(x^k)

S=sylvesterMatrix[d,e,x]//MatrixForm

Det[%]

Resultant[d,e,x]