Explanation

(b)
Since, $\alpha\ and\ \beta$ are the roots of the equation
$2x^{2} + 2(a + b) + a^{2} + b^{2} = 0$.
$\therefore(\alpha + \beta)^{2} = (a + b)^{2}$ and $\alpha\beta = \frac{a^{2}{+ b}^{2}}{2}$
Now, $(\alpha - \beta)^{2} = (\alpha + \beta)^{2} - 4\alpha\beta$
$= (a + b)^{2} - 4\left( \ \frac{a^{2}{+ b}^{2}}{2} \right)$
$= - (a - b)^{2}$
Now, the required equation whose roots are $(\alpha + \beta)^{2}$ and $(\alpha - \beta)^{2}$ is
$x^{2} - \{\ (\alpha + \beta)^{2} + (\alpha - \beta)^{2}\} x + (\alpha + \beta)^{2}(\alpha - \beta)^{2} = 0$
$\Rightarrow x^{2} - \left\{ (a + b)^{2} - (a - b)^{2} \right\} x - (a + b)^{2}(a - b)^{2} = 0$
$\Rightarrow x^{2} - 4abx - \left( a^{2} - b^{2} \right)^{2} = 0$