Example 8 Equation
$$
x^{3}-\sqrt{3} x^{2}-(2 \sqrt{3}+1) x+3+\sqrt{3}=0
$$

The three roots of the equation are $\qquad$
Let $\sqrt{3}=a$.
Transform the main variable to get
$$
\begin{array}{l}
a^{2}-\left(x^{2}+2 x-1\right) a+x^{3}-x=0 \\
\Rightarrow[a-(x-1)]\left[a-\left(x^{2}+x\right)\right]=0 \\
\Rightarrow x=a+1=\sqrt{3}+1 \text { or } x^{2}+x-\sqrt{3}=0 .
\end{array}
$$

Therefore, the solutions to the original equation are
$$
\sqrt{3}+1, \frac{-1 \pm \sqrt{1+4 \sqrt{3}}}{2}
$$