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

Resultant[a, b, x]

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]

res = Resultant[(((x - y))^2) - 2, (y^3) - 5, y]

Factor[res]

res = resultant[(((x - y))^2) - 2, (y^3) - 5, y]

Together[res]

?SylvesterMatrix

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

a = (∑_(k = 0)^7)(((k + 1))^3) (x^k)

b = (∑_(k = 0)^5) (k^2) (x^k)

S = SylvesterMatrix[a, b, x]

Det[S]

Resultant[a, b, x]

r = PolynomialRemainder[a, b, x]

T = 
      S /. {512 -> 0, 343 -> 0, 216 -> 0, 
          125 -> (368198/15625),  64 -> (518652/15625), 
          27 -> (357612/15625),  8 -> (111328/15625)}

Det[T]