Let $\alpha ,\beta$ are the roots of the quadratic equation $2x^2-5x+1=0$. If $S_n={\left(\alpha \right)}^{2n}+{\left(\beta \right)}^{2n}$, then  find the value of  $\frac{4S_{2021}+S_{2019}}{S_{2020}}$

${\alpha }^2,{\beta }^2$ are Roots of  $4x^2-21x+1=0$
Let  $c={\alpha }^2,d={\beta }^2\therefore s_n=c^n+d^n$
$\therefore 4S_{2021}+S_{2019}=4(c^{2021}+d^{2021})+(c^{2019}+d^{2019})$
$=c^{2019}(4c^2+1)+d^{2019}(4d^2+1)$
$=c^{2019}\left(21c\right)+d^{2019}\left(21d\right)$
$=21S_{2020}$