6・80 Let two complex numbers $x, y$, the sum of their squares is 7, and the sum of their cubes is 10. What is the largest possible real value of $x+y$?
[Solution] From the given conditions,
$$
\left\{\begin{array}{l}
x^{2}+y^{2}=7 \\
x^{3}+y^{3}=10
\end{array}\right.
$$

Let $x+y=t$. Then we have
$$
\begin{array}{l}
(x+y)^{3}=\left(x^{3}+y^{3}\right)+3 x y(x+y), \\
(x+y)^{3}=\left(x^{3}+y^{3}\right)+3(x+y) \cdot \frac{(x+y)^{2}-\left(x^{2}+y^{2}\right)}{2}
\end{array}
$$

Thus, $t^{3}=10+3 t \cdot \frac{t^{2}-7}{2}$,
which simplifies to $t^{3}-21 t+20=0$.
Therefore, we get $t_{1}=1, t_{2}=4, t_{3}=-5$. Hence, the largest possible real value of $x+y$ is 4.