Example 1. Solve the equation $8 x^{3}+8 \sqrt{3} x^{2}+6 x+\sqrt{3}$ $-1=0$
Let $\sqrt{3}=t$, the original equation becomes
$$
2 x t^{2}+\left(8 x^{2}+1\right) t+8 x^{3}-1=0 \text {. }
$$

Solving the quadratic equation for $t$ yields
$$
t_{1}=-2 x+1, t_{2}=\frac{4 x^{2}+2 x+1}{-2 x} .
$$

Given $t=\sqrt{3}$, the original equation's roots are
$$
\begin{array}{l}
x_{1}=\frac{1-\sqrt{3}}{2}, \\
x_{2}, 3=-\frac{1+\sqrt{3} \pm \sqrt[4]{12}}{4} .
\end{array}
$$