sum and product of roots

3α and 3β are the roots of x^2 + px + q = 0

3(α+β) = -p and 9αβ = q

So,

(α + 2β) + (β + 2α) = 3(α+β) = -p

(α + 2β)(β + 2α)

= 2(α^2 + β^2) + 5αβ

= 2(α+β)^2 + αβ

= 2p^2/9 + q/9

So, the quadratic equation in x whose roots are α+2β and β+2α is

x^2 + px + 2p^2/9 + q/9 = 0 or

9x^2 + 9px + (2p^2 + q) = 0