Given $\left|\begin{array}{ccc}x+\alpha & \beta & \gamma \\ \gamma & x+\beta & \alpha \\ \alpha & \beta & x+\gamma\end{array}\right|=0$
Operate $\mathrm{C}_{1} \rightarrow \mathrm{C}_{1}+\mathrm{C}_{2}+\mathrm{C}_{3}$
$\begin{array}{l}
\left|\begin{array}{ccc}
x+\alpha+\beta+\gamma & \beta & \gamma \\
x+\alpha+\beta+\gamma & x+\beta & \alpha \\
x+\alpha+\beta+\gamma & \beta & x+\gamma
\end{array}\right|=0 \\
=(x+\alpha+\beta+\gamma)\left|\begin{array}{ccc}
1 & \beta & \gamma \\
1 & x+\beta & \alpha \\
1 & \beta & x+\gamma
\end{array}\right|=0 \\
\Rightarrow x+\alpha+\beta+\gamma=0 \Rightarrow x=-(\alpha+\beta+\gamma)
\end{array}$
Again if
$$
\begin{array}{l}
\left|\begin{array}{ccc}
1 & \beta & \gamma \\
1 & x+\beta & \alpha \\
1 & \beta & \gamma
\end{array}\right|=0 \Rightarrow\left|\begin{array}{ccc}
1 & \beta & \gamma \\
0 & x & \alpha-\gamma \\
0 & 0 & x
\end{array}\right|=0 \\
\Rightarrow x^{2}=0 \Rightarrow x=0
\end{array}
$$
$\therefore$ Solutions of the equation are $x=0$,
$-(\alpha+\beta+\gamma)$