If $\alpha ,\beta ,\gamma$ are roots of $x^3+4x+1=0$, then $(\alpha +\beta {)}^{-1}+(\beta +\gamma {)}^{-1}+(\gamma +\alpha {)}^{-1}$ equals

The correct option is C $4$
We know that If $\alpha ,\beta ,\gamma$ are roots of $x^3+a{x}^2+bx+c=0$, then

$(\alpha +\beta {)}^{-1}+(\beta +\gamma {)}^{-1}+(\gamma +\alpha {)}^{-1}=\frac{a^2+b}{c-ab}$
Given $\alpha ,\beta ,\gamma$ are the roots of $x^3+4x+1=0$
Here, $a=0,b=4,c=1$
$\Rightarrow (\alpha +\beta {)}^{-1}+(\beta +\gamma {)}^{-1}+(\gamma +\alpha {)}^{-1}=\frac{a^2+b}{c-ab}=\frac{4}{1}$