Using the Polynomial roots formula,
x^2 coefficient = 1 Roots (p+q) = A Roots (pq) = B x^2 coefficient = 1 Roots (r+s) = B Roots (rs) = A
it is easy to derive p=q and r=s; then 2p=r^2, 2r=p^2, p=q=r=s=2, {A,B} ={4,4} as the sole solution in the positive integers.
(Other solutions not compliant with the terms of the puzzle are also possible.)
blackjack
flooble's webmaster puzzle