2020 AMC 10A Problem 14

Let $S_k=x^k+y^k$. Since $x+y=4$ and $xy=-2$, the numbers $x$ and $y$ satisfy $t^2-4t-2=0$, so $S_k=4S_{k-1}+2S_{k-2}$. Using $S_0=2$ and $S_1=4$, we get $S_2=20$, $S_3=88$, $S_4=392$, and $S_5=1744$. The expression is $x+y+\dfrac{x^5+y^5}{x^2y^2}$ $=4+\dfrac{1744}{4}=440$. Thus, D is the correct answer.