Given equation is

\( \Large \left(5+\sqrt{2}\right)x^{2}- \left(4+\sqrt{5}\right) x+8 +2\sqrt{5} = 0 \)

Let \( \Large x_{1} \) and \( \Large x_{2} \) are the root of the equation.

=> \( \Large x_{1}+x_{2}=\frac{4+\sqrt{5}}{5+\sqrt{2}} \) ...(i)

and \( \Large x_{1}x_{2}=\frac{8+2\sqrt{5}}{5+\sqrt{2}}=\frac{2 \left(4+\sqrt{5}\right) }{5+\sqrt{2}}=2 \left(x_{1}+x_{2}\right) \) ...(ii)

Harmonic mean = \( \Large \frac{2x_{1}x_{2}}{x_{1}+x_{2}}=\frac{4 \left(x_{1}+x_{2}\right) }{ \left(x_{1}+x_{2}\right) }=4 \)