Explanation

(b)
Given equation is
$(a + 1)x^{2} - (a + 2)x + (a + 3) = 0$
Since, roots are equal in magnitude and opposite in sign
$\therefore$ Coefficient of $x$ is zero $ie,\ a + 2 = 0$
$\Rightarrow \ \ a = - 2$ ...(i)
$\therefore$ Equation is
$( - 2 + 1)x^{2} - ( - 2 + 2)x + ( - 2 + 3) = 0$
$\Rightarrow \ \ \ \ \ \ - x^{2} + 1 = 0$
$\Rightarrow \ \ \ x = \pm 1$ ...(ii)
Only option (b) $i.e.,\ \pm \frac{1}{2}a$ satisfies Eqs. (i) and (ii)