Let p and q be real numbers such that p≠0, p³≠q and p³≠-q. If α and β are non zero complex numbers satisfying α+β=-p and α³+β³=q, then a quadratic equation having αβ and βα as its roots is (p³+q)x²- (p³+2q)x + (p³+q)=0

Let the new equation be x²+Bx+C=0 where, A=1 is the coefficient of x², B is the coefficient of x, and C is the constant term. We know that, for a quadratic equation ax²+bx+c=0, Sum of the roots = -b/a and Product of roots = c/a
∴ B = -(αβ+βα) and C = αβ.βα
x² - (αβ+βα)x + αβ.βα = 0
x² - (α²+β²)x + 1 = 0
x² - ( (α+β)² - 2αβ)x + 1 = 0
Now, we have α³+β³=q
(α+β)³ - 3αβ(α+β) = q
-p³ + 3pαβ = q
αβ = (q+p³)/3p
x² - (p² - 2(q+p³)/3p)x + 1 = 0
(p³+q)x² - (3p³-2(q+p³))x + (p³+q) = 0
(p³+q)x² - (p³+2q)x + (p³+q) = 0