Question: Why doesn't factor work?

separable form should be (x - something in terms of x1,x2,x3,x4)*(x- something in terms of x1,x2,x3,x4)*(x -something in terms of x1,x2,x3,x4 )*(x - something in terms of x1,x2,x3,x4)

factor(f, x);
Error, (in factor) 2nd argument, x, is not a valid algebraic extension

how to factor into separable form?

ferrai := -x1^3*x2*x3*x4-x1^2*x2^2*x3^2-x1^2*x2^2*x4^2-x1^2*x3^2*x4^2-x1*x2^3*x3*x4-x1*x2*x3^3*x4-x1*x2*x3*x4^3-x2^2*x3^2*x4^2+x1^2*x2*x3*y+x1^2*x2*x4*y+x1^2*x3*x4*y+x1*x2^2*x3*y+x1*x2^2*x4*y+x1*x2*x3^2*y+x1*x2*x4^2*y+x1*x3^2*x4*y+x1*x3*x4^2*y+x2^2*x3*x4*y+x2*x3^2*x4*y+x2*x3*x4^2*y-x1*x2*y^2-x1*x3*y^2-x1*x4*y^2-x2*x3*y^2-x2*x4*y^2-x3*x4*y^2+y^3;
coeff(ferrai, y^3);
coeff(ferrai, y^2);
coeff(ferrai, y);
res := simplify(ferrai - coeff(ferrai, y^3)*y^3 - coeff(ferrai, y^2)*y^2 - coeff(ferrai, y)*y);

c2 := -coeff(ferrai, y^2)/coeff(ferrai, y^3);
sys1 := c1*c3 - 4*c4 = coeff(ferrai, y)/coeff(ferrai, y^3);
sys2 := -c3^2-(c1^2)*c4+4*c2*c4 = res/coeff(ferrai, y^3);

diff(lhs(sys1),c1) = diff(rhs(sys1),c1);
diff(lhs(sys1),c3) = diff(rhs(sys1),c3);
diff(lhs(sys1),c4) = diff(rhs(sys1),c4);

diff(lhs(sys2),c1) = diff(rhs(sys2),c1);
diff(lhs(sys2),c3) = diff(rhs(sys2),c3);
diff(lhs(sys2),c4) = diff(rhs(sys2),c4);

c1sol := solve(diff(lhs(sys2),c4) = diff(rhs(sys2),c4), c1)[1];
indets(subs(c1=c1sol,sys1));
indets(subs(c1=c1sol,sys2));
indets([subs(c1 = c1sol, sys1), subs(c1 = c1sol, sys2)]);
result := solve({subs(c1 = c1sol, sys1), subs(c1 = c1sol, sys2)},{c3,c4}, explicit);

c1sol := solve(diff(lhs(sys2),c4) = diff(rhs(sys2),c4), c1)[1];
c2sol := -coeff(ferrai, y^2)/coeff(ferrai, y^3);
c3sol := rhs(result[1][1]);
c4sol := rhs(result[1][2]);
f := x^4-c1sol*x^3+c2sol*x^2-c3sol*x+c4sol;
aa := factor(f);
aa := factor(f,x);