Answer: \(x^2 - 4pqx - (p^2 - q^2)^2 = 0\)
Let \(\alpha, \beta\) be the roots of \(2x^2 + 2(p + q)x + p^2 + q^2 = 0\).
\[\alpha + \beta = -(p + q)\]\[\alpha\beta = \frac{p^2 + q^2}{2}\]Roots of the required equation: \(m = (\alpha + \beta)^2 = (p + q)^2\) and
\[n = (\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta\]\[= (p + q)^2 - 2(p^2 + q^2)\]\[= -(p - q)^2\]Sum \(m + n = (p + q)^2 - (p - q)^2 = 4pq\).
Product \(mn = -(p + q)^2(p - q)^2 = -(p^2 - q^2)^2\).
Required equation: \(x^2 - 4pqx - (p^2 - q^2)^2 = 0\).