Explanation

(c)
Given, $\alpha,\ \beta\$and$\ \gamma$ are the roots of $x^{3} + 4x + 1 = 0$
$\therefore\ \ \alpha + \beta + \gamma = 0,\ \alpha\beta + \beta\gamma + \gamma\alpha = 4,\ \alpha\beta\gamma = - 1$
Now, $\frac{\ \alpha^{2}}{\beta + \gamma} + \frac{\beta^{2}}{\gamma + \alpha} + \frac{\gamma^{2}}{\alpha + \beta} = \frac{\alpha^{2}}{- \alpha} + \frac{\beta^{2}}{- \beta} + \frac{\gamma^{2}}{- \gamma}$
$= - (\alpha + \beta + \gamma) = 0$
$\frac{\alpha^{2}\beta^{2}}{(\beta + \gamma)(\gamma + \alpha)} + \frac{\beta^{2}\gamma^{2}}{(\gamma + \alpha)(\alpha + \beta)} + \frac{\gamma^{2}\alpha^{2}}{(\beta + \gamma)(\alpha + \beta)}$
$= \alpha\beta + \beta\gamma + \gamma\alpha = 4$
And $\frac{\alpha^{2}\beta^{2}\gamma^{2}}{(\beta + \gamma)(\gamma + \alpha)(\alpha + \beta)} = - \alpha\beta\gamma = 1\ \ \ (\because\alpha + \beta + \gamma = 0)$
$\therefore$ Required equation is
$x^{3} + 4x - 1 = 0$