Explanation

(b)
Let all four roots are imaginary. Then roots of both equation $P(x) = 0\ and\ Q(x) = 0$ are imaginary.
Thus, $b^{2} - 4ac < 0;d^{2} - 4ac < 0,\ so\ b^{2} + d^{2} < 0\ which\ is\ impossible\ unless\ b = 0,d = 0$.
So, if $b \neq 0\ or\ d \neq 0$ at least two roots must be real, if $b = 0,\ d = 0$ we have the equations
$P(x) = ax^{2} + c = 0$
and $Q(x) = - ax^{2} + c = 0$
or $x^{2} = - \frac{c}{a};x^{2} = \frac{c}{a}\ as\ one\ of\ \frac{c}{a}and - \frac{c}{a}$ must be positive so two roots must be real.