If $\alpha, \beta$ are roots of the equation $x^2+5(\sqrt{2}) x+10=0, \alpha>\beta$ and $P_n=\alpha^n-\beta^n$ for each positive integer $n$, then the value of $\left(\frac{\mathrm{P}_{17} \mathrm{P}_{20}+5 \sqrt{2} \mathrm{P}_{17} \mathrm{P}_{19}}{\mathrm{P}_{17} \mathrm{P}_{20}+5 \sqrt{2} \mathrm{P}_{18}^2}\right)$ is equal to $\_\_\_\_$ .