Explanation

(b)
Given equation is $\left( 5 + \sqrt{2} \right)x^{2} - \left( 4 + \sqrt{5} \right)x + 8 + 2\sqrt{5} = 0$.
Let $x_{1}\ and\ x_{2}$ are the roots of the equation.
$\Rightarrow x_{1} + x_{2} = \frac{4 + \sqrt{5}}{5 + \sqrt{2}}$ ….(i)
and $x_{1}x_{2} = \frac{8 + 2\sqrt{5}}{5 + \sqrt{2}} = \frac{2\left( 4 + \sqrt{5} \right)}{5 + \sqrt{2}} = 2(x_{1} + x_{2})$ ...(ii)
$\therefore$ Harmonic mean $= \frac{2x_{1}x_{2}}{x_{1} + x_{2}} = \frac{4(x_{1} + x_{2})}{(x_{1} + x_{2})} = 4$ [from Eq. (ii)]