Quadratic Equation And Inequalities

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 \(\left( {{{{P_{17}}{P_{20}} + 5\sqrt 2 {P_{17}}{P_{19}}} \over {{P_{18}}{P_{19}} + 5\sqrt 2 P_{18}^2}}} \right)\) is equal to _________.