Explanation

(a)
$\left( x + \frac{1}{x} \right)^{3} + \left( x + \frac{1}{x} \right) = 0$
$\Rightarrow \ \ \left( x + \frac{1}{x} \right)\left\lbrack \left( x + \frac{1}{x} \right)^{2} + 1 \right\rbrack = 0$
$\Rightarrow x + \frac{1}{x} = 0$
$\Rightarrow x^{2} = - 1$ which is not possible
Hence, no real roots exist