SetDirectory[NotebookDirectory[]]

Helper methods

<< Bezout.m

x[coef_, var_, exp_] := Plus @@ Table[(Subscript[coef, j])(var^j), {j, 0, exp}]

BezoutDet = (Subscript[v, i_,j_]) :>xs[[i+1]] ys[[j+1]] - xs[[j+1]] ys[[i +1]]

Define some parametric curves

x1 = x[cx1, t, 2]

y1 = x[cy1, t, 2]

x2 = x[cx2, s, 2]

y2 = x[cy2, s, 2]

Set axes equal to one another, grouped by ‘t’

xs = CoefficientList[x2 - x1, t]

ys = CoefficientList[y2 - y1, t]

Compare Resultant (Sylvester?) to Bezout

Resultant[x2 - x1, y2 - y1, t]

Collect[Expand[BezoutMatrix[2] /. BezoutDet], s] //MatrixForm