Explanation

(b)
Here, $\alpha + \beta = \frac{- b}{a},\ \alpha\beta = \frac{c}{a}$
So, $\left( 1 + \alpha + \alpha^{2} \right)(1 + \beta + \beta^{2})$
$= 1 + \beta + \beta^{2} + \alpha + \alpha\beta + \alpha\beta^{2} + \alpha^{2} + \alpha^{2}\beta + \alpha^{2}\beta^{2}$
$= 1 + (\alpha + \beta) + \alpha\beta + \alpha\beta(\alpha + \beta) + (\alpha\beta)^{2} + \alpha^{2} + \beta^{2}$
$= 1 + (\alpha + \beta) - \alpha\beta + \alpha\beta(\alpha + \beta) + (\alpha\beta)^{2} + (\alpha + \beta)^{2}$
$= 1 - \frac{b}{a} - \frac{c}{a} - \frac{bc}{a^{2}} + \frac{c^{2}}{a^{2}} + \frac{b^{2}}{a^{2}}$
$= \frac{a^{2} + b^{2} + c^{2} - ab - bc - ca}{a^{2}}$
$= \frac{1}{2a^{2}}\left( 2a^{2} + 2b^{2} + 2c^{2} - 2ab - 2bc - 2ca \right)$
$= \frac{1}{2a^{2}}\lbrack(a - b)^{2} + (b - c)^{2} + (c - a)^{2}\rbrack \geq 0$