Explanation

(d)
Here, $\alpha + \beta + \gamma = 0,\ \alpha\beta + \beta\gamma + \gamma\alpha = - 8,\ \alpha\beta\gamma = - 8$ ...(i)
$\therefore(\alpha + \beta + \gamma)^{2} = 0$
$\Rightarrow \ \alpha^{2} + \beta^{2} + \gamma^{2} + 2(\alpha\beta + \beta\gamma + \gamma\alpha) = 0$
$\Rightarrow \ \sum_{}^{}{\alpha^{2} = - 2( - 8) = 16}$ [from Eq. (i)]
And $\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} = \frac{\alpha + \beta + \gamma}{\alpha\beta\gamma}$
$\Rightarrow \ \frac{1}{\sum_{}^{}{\alpha\beta}} = \frac{0}{- 8} = 0$ [from Eq. (i)]