4・128 Find the relationship between the coefficients $a, b, c$ for the system of equations
$$\left\{\begin{array}{l}
a x^{2}+b x+c=0, \\
b x^{2}+c x+a=0, \\
c x^{2}+a x+b=0 .
\end{array}\right.$$

to have real solutions.
[Solution] Adding the equations in the system, we get
$$(a+b+c) x^{2}+(a+b+c) x+(a+b+c)=0$$

Thus, $(a+b+c)\left(x^{2}+x+1\right)=0$.
Since for any real number $x, x^{2}+x+1 \neq 0$, the necessary condition for the system of equations to have a solution is $a+b+c=0$.

Conversely, if $a+b+c=0$, then $x=1$ is clearly a solution to the system of equations.
Therefore, $a+b+c=0$ is also a sufficient condition for the system of equations to have a solution.