sage: var('a','b','c','d','x','y','z','a1','a2','a3','a4','b1','b2','b3','p','q')
sage: R.<u>=QQ[]
sage: f=(u-a)*(u-b)*(u-c)*(u-d)
sage: f
a*b*c*d - a*b*c*u - a*b*d*u - a*c*d*u - b*c*d*u + a*b*u^2 + a*c*u^2 + b*c*u^2 + a*d*u^2 + b*d*u^2 + c*d*u^2 - a*u^3 - b*u^3 - c*u^3 - d*u^3 + u^4
sage: a1=f.coefficient(u^3)
sage: a1
-a - b - c - d
sage: a2=f.coefficient(u^2)
sage: a2
a*b + a*c + b*c + a*d + b*d + c*d
sage: a3=f.coefficient(u^1)
sage: a4=a*b*c*d
sage: b1=-a2
sage: b2=a1*a3-4*a4
sage: b3=4*a2*a4-a3^2-a1^2*a4
sage: p=-b1^2/3+b2
sage: q=-b1*b2/3+b3+2/27*b1^3
sage: p
-4*a*b*c*d - 1/3*(a*b + a*c + b*c + a*d + b*d + c*d)^2 + (a*b*c + a*b*d + a*c*d + b*c*d)*(a + b + c + d)
sage: q
-(a + b + c + d)^2*a*b*c*d + 4*(a*b + a*c + b*c + a*d + b*d + c*d)*a*b*c*d - 2/27*(a*b + a*c + b*c + a*d + b*d + c*d)^3 - (a*b*c + a*b*d + a*c*d + b*c*d)^2 - 1/3*(4*a*b*c*d - (a*b*c + a*b*d + a*c*d + b*c*d)*(a + b + c + d))*(a*b + a*c + b*c + a*d + b*d + c*d)
sage: expand(-4*p^3-27*q^2)
a^6*b^4*c^2 - 2*a^5*b^5*c^2 + a^4*b^6*c^2 - 2*a^6*b^3*c^3 + 2*a^5*b^4*c^3 + 2*a^4*b^5*c^3 - 2*a^3*b^6*c^3 + a^6*b^2*c^4 + 2*a^5*b^3*c^4 - 6*a^4*b^4*c^4 + 2*a^3*b^5*c^4 + a^2*b^6*c^4 - 2*a^5*b^2*c^5 + 2*a^4*b^3*c^5 + 2*a^3*b^4*c^5 - 2*a^2*b^5*c^5 + a^4*b^2*c^6 - 2*a^3*b^3*c^6 + a^2*b^4*c^6 - 2*a^6*b^4*c*d + 4*a^5*b^5*c*d - 2*a^4*b^6*c*d + 2*a^6*b^3*c^2*d - 2*a^5*b^4*c^2*d - 2*a^4*b^5*c^2*d + 2*a^3*b^6*c^2*d + 2*a^6*b^2*c^3*d - 4*a^5*b^3*c^3*d + 4*a^4*b^4*c^3*d - 4*a^3*b^5*c^3*d + 2*a^2*b^6*c^3*d - 2*a^6*b*c^4*d - 2*a^5*b^2*c^4*d + 4*a^4*b^3*c^4*d + 4*a^3*b^4*c^4*d - 2*a^2*b^5*c^4*d - 2*a*b^6*c^4*d + 4*a^5*b*c^5*d - 2*a^4*b^2*c^5*d - 4*a^3*b^3*c^5*d - 2*a^2*b^4*c^5*d + 4*a*b^5*c^5*d - 2*a^4*b*c^6*d + 2*a^3*b^2*c^6*d + 2*a^2*b^3*c^6*d - 2*a*b^4*c^6*d + a^6*b^4*d^2 - 2*a^5*b^5*d^2 + a^4*b^6*d^2 + 2*a^6*b^3*c*d^2 - 2*a^5*b^4*c*d^2 - 2*a^4*b^5*c*d^2 + 2*a^3*b^6*c*d^2 - 6*a^6*b^2*c^2*d^2 + 4*a^5*b^3*c^2*d^2 + 4*a^4*b^4*c^2*d^2 + 4*a^3*b^5*c^2*d^2 - 6*a^2*b^6*c^2*d^2 + 2*a^6*b*c^3*d^2 + 4*a^5*b^2*c^3*d^2 - 6*a^4*b^3*c^3*d^2 - 6*a^3*b^4*c^3*d^2 + 4*a^2*b^5*c^3*d^2 + 2*a*b^6*c^3*d^2 + a^6*c^4*d^2 - 2*a^5*b*c^4*d^2 + 4*a^4*b^2*c^4*d^2 - 6*a^3*b^3*c^4*d^2 + 4*a^2*b^4*c^4*d^2 - 2*a*b^5*c^4*d^2 + b^6*c^4*d^2 - 2*a^5*c^5*d^2 - 2*a^4*b*c^5*d^2 + 4*a^3*b^2*c^5*d^2 + 4*a^2*b^3*c^5*d^2 - 2*a*b^4*c^5*d^2 - 2*b^5*c^5*d^2 + a^4*c^6*d^2 + 2*a^3*b*c^6*d^2 - 6*a^2*b^2*c^6*d^2 + 2*a*b^3*c^6*d^2 + b^4*c^6*d^2 - 2*a^6*b^3*d^3 + 2*a^5*b^4*d^3 + 2*a^4*b^5*d^3 - 2*a^3*b^6*d^3 + 2*a^6*b^2*c*d^3 - 4*a^5*b^3*c*d^3 + 4*a^4*b^4*c*d^3 - 4*a^3*b^5*c*d^3 + 2*a^2*b^6*c*d^3 + 2*a^6*b*c^2*d^3 + 4*a^5*b^2*c^2*d^3 - 6*a^4*b^3*c^2*d^3 - 6*a^3*b^4*c^2*d^3 + 4*a^2*b^5*c^2*d^3 + 2*a*b^6*c^2*d^3 - 2*a^6*c^3*d^3 - 4*a^5*b*c^3*d^3 - 6*a^4*b^2*c^3*d^3 + 24*a^3*b^3*c^3*d^3 - 6*a^2*b^4*c^3*d^3 - 4*a*b^5*c^3*d^3 - 2*b^6*c^3*d^3 + 2*a^5*c^4*d^3 + 4*a^4*b*c^4*d^3 - 6*a^3*b^2*c^4*d^3 - 6*a^2*b^3*c^4*d^3 + 4*a*b^4*c^4*d^3 + 2*b^5*c^4*d^3 + 2*a^4*c^5*d^3 - 4*a^3*b*c^5*d^3 + 4*a^2*b^2*c^5*d^3 - 4*a*b^3*c^5*d^3 + 2*b^4*c^5*d^3 - 2*a^3*c^6*d^3 + 2*a^2*b*c^6*d^3 + 2*a*b^2*c^6*d^3 - 2*b^3*c^6*d^3 + a^6*b^2*d^4 + 2*a^5*b^3*d^4 - 6*a^4*b^4*d^4 + 2*a^3*b^5*d^4 + a^2*b^6*d^4 - 2*a^6*b*c*d^4 - 2*a^5*b^2*c*d^4 + 4*a^4*b^3*c*d^4 + 4*a^3*b^4*c*d^4 - 2*a^2*b^5*c*d^4 - 2*a*b^6*c*d^4 + a^6*c^2*d^4 - 2*a^5*b*c^2*d^4 + 4*a^4*b^2*c^2*d^4 - 6*a^3*b^3*c^2*d^4 + 4*a^2*b^4*c^2*d^4 - 2*a*b^5*c^2*d^4 + b^6*c^2*d^4 + 2*a^5*c^3*d^4 + 4*a^4*b*c^3*d^4 - 6*a^3*b^2*c^3*d^4 - 6*a^2*b^3*c^3*d^4 + 4*a*b^4*c^3*d^4 + 2*b^5*c^3*d^4 - 6*a^4*c^4*d^4 + 4*a^3*b*c^4*d^4 + 4*a^2*b^2*c^4*d^4 + 4*a*b^3*c^4*d^4 - 6*b^4*c^4*d^4 + 2*a^3*c^5*d^4 - 2*a^2*b*c^5*d^4 - 2*a*b^2*c^5*d^4 + 2*b^3*c^5*d^4 + a^2*c^6*d^4 - 2*a*b*c^6*d^4 + b^2*c^6*d^4 - 2*a^5*b^2*d^5 + 2*a^4*b^3*d^5 + 2*a^3*b^4*d^5 - 2*a^2*b^5*d^5 + 4*a^5*b*c*d^5 - 2*a^4*b^2*c*d^5 - 4*a^3*b^3*c*d^5 - 2*a^2*b^4*c*d^5 + 4*a*b^5*c*d^5 - 2*a^5*c^2*d^5 - 2*a^4*b*c^2*d^5 + 4*a^3*b^2*c^2*d^5 + 4*a^2*b^3*c^2*d^5 - 2*a*b^4*c^2*d^5 - 2*b^5*c^2*d^5 + 2*a^4*c^3*d^5 - 4*a^3*b*c^3*d^5 + 4*a^2*b^2*c^3*d^5 - 4*a*b^3*c^3*d^5 + 2*b^4*c^3*d^5 + 2*a^3*c^4*d^5 - 2*a^2*b*c^4*d^5 - 2*a*b^2*c^4*d^5 + 2*b^3*c^4*d^5 - 2*a^2*c^5*d^5 + 4*a*b*c^5*d^5 - 2*b^2*c^5*d^5 + a^4*b^2*d^6 - 2*a^3*b^3*d^6 + a^2*b^4*d^6 - 2*a^4*b*c*d^6 + 2*a^3*b^2*c*d^6 + 2*a^2*b^3*c*d^6 - 2*a*b^4*c*d^6 + a^4*c^2*d^6 + 2*a^3*b*c^2*d^6 - 6*a^2*b^2*c^2*d^6 + 2*a*b^3*c^2*d^6 + b^4*c^2*d^6 - 2*a^3*c^3*d^6 + 2*a^2*b*c^3*d^6 + 2*a*b^2*c^3*d^6 - 2*b^3*c^3*d^6 + a^2*c^4*d^6 - 2*a*b*c^4*d^6 + b^2*c^4*d^6
sage: expand(((a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d))^2)
a^6*b^4*c^2 - 2*a^5*b^5*c^2 + a^4*b^6*c^2 - 2*a^6*b^3*c^3 + 2*a^5*b^4*c^3 + 2*a^4*b^5*c^3 - 2*a^3*b^6*c^3 + a^6*b^2*c^4 + 2*a^5*b^3*c^4 - 6*a^4*b^4*c^4 + 2*a^3*b^5*c^4 + a^2*b^6*c^4 - 2*a^5*b^2*c^5 + 2*a^4*b^3*c^5 + 2*a^3*b^4*c^5 - 2*a^2*b^5*c^5 + a^4*b^2*c^6 - 2*a^3*b^3*c^6 + a^2*b^4*c^6 - 2*a^6*b^4*c*d + 4*a^5*b^5*c*d - 2*a^4*b^6*c*d + 2*a^6*b^3*c^2*d - 2*a^5*b^4*c^2*d - 2*a^4*b^5*c^2*d + 2*a^3*b^6*c^2*d + 2*a^6*b^2*c^3*d - 4*a^5*b^3*c^3*d + 4*a^4*b^4*c^3*d - 4*a^3*b^5*c^3*d + 2*a^2*b^6*c^3*d - 2*a^6*b*c^4*d - 2*a^5*b^2*c^4*d + 4*a^4*b^3*c^4*d + 4*a^3*b^4*c^4*d - 2*a^2*b^5*c^4*d - 2*a*b^6*c^4*d + 4*a^5*b*c^5*d - 2*a^4*b^2*c^5*d - 4*a^3*b^3*c^5*d - 2*a^2*b^4*c^5*d + 4*a*b^5*c^5*d - 2*a^4*b*c^6*d + 2*a^3*b^2*c^6*d + 2*a^2*b^3*c^6*d - 2*a*b^4*c^6*d + a^6*b^4*d^2 - 2*a^5*b^5*d^2 + a^4*b^6*d^2 + 2*a^6*b^3*c*d^2 - 2*a^5*b^4*c*d^2 - 2*a^4*b^5*c*d^2 + 2*a^3*b^6*c*d^2 - 6*a^6*b^2*c^2*d^2 + 4*a^5*b^3*c^2*d^2 + 4*a^4*b^4*c^2*d^2 + 4*a^3*b^5*c^2*d^2 - 6*a^2*b^6*c^2*d^2 + 2*a^6*b*c^3*d^2 + 4*a^5*b^2*c^3*d^2 - 6*a^4*b^3*c^3*d^2 - 6*a^3*b^4*c^3*d^2 + 4*a^2*b^5*c^3*d^2 + 2*a*b^6*c^3*d^2 + a^6*c^4*d^2 - 2*a^5*b*c^4*d^2 + 4*a^4*b^2*c^4*d^2 - 6*a^3*b^3*c^4*d^2 + 4*a^2*b^4*c^4*d^2 - 2*a*b^5*c^4*d^2 + b^6*c^4*d^2 - 2*a^5*c^5*d^2 - 2*a^4*b*c^5*d^2 + 4*a^3*b^2*c^5*d^2 + 4*a^2*b^3*c^5*d^2 - 2*a*b^4*c^5*d^2 - 2*b^5*c^5*d^2 + a^4*c^6*d^2 + 2*a^3*b*c^6*d^2 - 6*a^2*b^2*c^6*d^2 + 2*a*b^3*c^6*d^2 + b^4*c^6*d^2 - 2*a^6*b^3*d^3 + 2*a^5*b^4*d^3 + 2*a^4*b^5*d^3 - 2*a^3*b^6*d^3 + 2*a^6*b^2*c*d^3 - 4*a^5*b^3*c*d^3 + 4*a^4*b^4*c*d^3 - 4*a^3*b^5*c*d^3 + 2*a^2*b^6*c*d^3 + 2*a^6*b*c^2*d^3 + 4*a^5*b^2*c^2*d^3 - 6*a^4*b^3*c^2*d^3 - 6*a^3*b^4*c^2*d^3 + 4*a^2*b^5*c^2*d^3 + 2*a*b^6*c^2*d^3 - 2*a^6*c^3*d^3 - 4*a^5*b*c^3*d^3 - 6*a^4*b^2*c^3*d^3 + 24*a^3*b^3*c^3*d^3 - 6*a^2*b^4*c^3*d^3 - 4*a*b^5*c^3*d^3 - 2*b^6*c^3*d^3 + 2*a^5*c^4*d^3 + 4*a^4*b*c^4*d^3 - 6*a^3*b^2*c^4*d^3 - 6*a^2*b^3*c^4*d^3 + 4*a*b^4*c^4*d^3 + 2*b^5*c^4*d^3 + 2*a^4*c^5*d^3 - 4*a^3*b*c^5*d^3 + 4*a^2*b^2*c^5*d^3 - 4*a*b^3*c^5*d^3 + 2*b^4*c^5*d^3 - 2*a^3*c^6*d^3 + 2*a^2*b*c^6*d^3 + 2*a*b^2*c^6*d^3 - 2*b^3*c^6*d^3 + a^6*b^2*d^4 + 2*a^5*b^3*d^4 - 6*a^4*b^4*d^4 + 2*a^3*b^5*d^4 + a^2*b^6*d^4 - 2*a^6*b*c*d^4 - 2*a^5*b^2*c*d^4 + 4*a^4*b^3*c*d^4 + 4*a^3*b^4*c*d^4 - 2*a^2*b^5*c*d^4 - 2*a*b^6*c*d^4 + a^6*c^2*d^4 - 2*a^5*b*c^2*d^4 + 4*a^4*b^2*c^2*d^4 - 6*a^3*b^3*c^2*d^4 + 4*a^2*b^4*c^2*d^4 - 2*a*b^5*c^2*d^4 + b^6*c^2*d^4 + 2*a^5*c^3*d^4 + 4*a^4*b*c^3*d^4 - 6*a^3*b^2*c^3*d^4 - 6*a^2*b^3*c^3*d^4 + 4*a*b^4*c^3*d^4 + 2*b^5*c^3*d^4 - 6*a^4*c^4*d^4 + 4*a^3*b*c^4*d^4 + 4*a^2*b^2*c^4*d^4 + 4*a*b^3*c^4*d^4 - 6*b^4*c^4*d^4 + 2*a^3*c^5*d^4 - 2*a^2*b*c^5*d^4 - 2*a*b^2*c^5*d^4 + 2*b^3*c^5*d^4 + a^2*c^6*d^4 - 2*a*b*c^6*d^4 + b^2*c^6*d^4 - 2*a^5*b^2*d^5 + 2*a^4*b^3*d^5 + 2*a^3*b^4*d^5 - 2*a^2*b^5*d^5 + 4*a^5*b*c*d^5 - 2*a^4*b^2*c*d^5 - 4*a^3*b^3*c*d^5 - 2*a^2*b^4*c*d^5 + 4*a*b^5*c*d^5 - 2*a^5*c^2*d^5 - 2*a^4*b*c^2*d^5 + 4*a^3*b^2*c^2*d^5 + 4*a^2*b^3*c^2*d^5 - 2*a*b^4*c^2*d^5 - 2*b^5*c^2*d^5 + 2*a^4*c^3*d^5 - 4*a^3*b*c^3*d^5 + 4*a^2*b^2*c^3*d^5 - 4*a*b^3*c^3*d^5 + 2*b^4*c^3*d^5 + 2*a^3*c^4*d^5 - 2*a^2*b*c^4*d^5 - 2*a*b^2*c^4*d^5 + 2*b^3*c^4*d^5 - 2*a^2*c^5*d^5 + 4*a*b*c^5*d^5 - 2*b^2*c^5*d^5 + a^4*b^2*d^6 - 2*a^3*b^3*d^6 + a^2*b^4*d^6 - 2*a^4*b*c*d^6 + 2*a^3*b^2*c*d^6 + 2*a^2*b^3*c*d^6 - 2*a*b^4*c*d^6 + a^4*c^2*d^6 + 2*a^3*b*c^2*d^6 - 6*a^2*b^2*c^2*d^6 + 2*a*b^3*c^2*d^6 + b^4*c^2*d^6 - 2*a^3*c^3*d^6 + 2*a^2*b*c^3*d^6 + 2*a*b^2*c^3*d^6 - 2*b^3*c^3*d^6 + a^2*c^4*d^6 - 2*a*b*c^4*d^6 + b^2*c^4*d^6
sage: A=expand(-4*p^3-27*q^2)
sage: B=expand(((a-b)*(a-c)*(a-d)*(b-c)*(b-d)*(c-d))^2)
sage: A-B
0