α, β are roots of x^2 + px + q = 0. Form an equation whose roots are (α + β)^2 and (α - β)^2.

Equation x^2 + px + q has roots α, β. By using the sum and product of roots formulae, we'll get

α + β = -p               (1)

αβ = q                    (2)

We have to find an equation who roots are (α + β)^2 and (α - β)^2.

We know that any Quadratic equation can be written as x^2 - (sum of its roots)x + product of its roots = 0. Hence to find the equation with roots (α + β)^2 and (α - β)^2, we'll have to find the sum and product of those roots.

Sum of roots

= (α + β)^2 + (α - β)^2

= (-p)^2 + α^2 + β^2 - 2αβ               (by Using (1))

= p^2 + (α + β)^2 - 2αβ - 2αβ        (by using a^2 + b^2 = (a + b)^2 - 2ab)

= p^2 + (α + β)^2 - 4αβ

= p^2 + (-p)^2 - 4αβ                         (by using (1))

= p^2 + p^2 - 4q                               (by using (2))

= 2p^2 - 4q

= 2(p^2 - 2q)

Product of roots

= (α + β)^2 * (α - β)^2

= (-p)^2 . (α^2 + β^2 - 2αβ)             (by using (1))

= p^2 . [(α + β)^2 - 2αβ - 2αβ]      (by using a^2 + b^2 = (a + b)^2 - 2ab)

= p^2 . [(α + β)^2 - 4αβ]

= p^2 . [(-p)^2 - 4]                            (by using (1) and (2))

= p^2 . (p^2 - 4)                         

= p^4 - 4p^2

We've found out that the sum and product of roots of the desired equation are (2p^2 - 4q) and (p^4 - 4p^2) respectively. So, we can say that the equation is

x^2 - (sum of roots)x + product of roots = 0

=> x^2 - 2(p^2 - 4q)x + (p^4 - 4p^2) = 0