AMC 10 2020a Problem 14

Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of $$x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?$$ $\textbf{(A)}\ 360 \textbf{(B)}\ 400 \textbf{(C)}\ 420 \textbf{(D)}\ 440 \textbf{(E)}\ 480$