Five, let the sum of the squares of two complex numbers x, $\mathrm{y}$ be 7, and their sum of cubes be 10. The largest real value that $x+y$ can take is what?

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Translation:
Five, let the sum of the squares of two complex numbers x, $\mathrm{y}$ be 7, and their sum of cubes be 10. The largest real value that $x+y$ can take is what?
$$
\begin{array}{l}
\because x^{2}+y^{2}=7, \\
\therefore x^{3}+y^{3}=10=(x+y)\left(x^{2}+y^{2}-x y\right) \\
=(x+y)\left[7-\frac{(x+y)^{2}-\left(x^{2}+y^{2}\right)}{2}\right] \\
=(x+y)\left[7-\frac{\left.(x+y)^{2}-7\right] .}{2}\right] .
\end{array}
$$

Let $x+y=t$ , then
$$
\begin{array}{c}
10=\frac{21 t-t^{3}}{2}, \\
t^{3}-21 t+20=0, \\
(t-1)(t-4)(t+5)=0 . \\
\therefore t_{1}=1, t_{2}=4, t_{3}=-5 .
\end{array}
$$

That is, the largest real value that $\mathrm{x}+\mathrm{y}$ can take is 4 .