Example 3 Given $(\sqrt{3}+1) x=\sqrt{3}-1$. Then $x^{4}-5 x^{3}+6 x^{2}-5 x+4=$ $\qquad$
(2013, Huanggang City, Hubei Province Mathematics Competition)
Solve: From $x=\frac{\sqrt{3}-1}{\sqrt{3}+1}=2-\sqrt{3}$, we get $x-2=\sqrt{3}$. Squaring it, we obtain
$$
x^{2}-4 x+1=0 \text {. }
$$

Use equation (1) to reduce the power of the target algebraic expression. Performing polynomial division, we get
$$
\begin{array}{l}
x^{4}-5 x^{3}+6 x^{2}-5 x+4 \\
=\left(x^{2}-4 x+1\right)\left(x^{2}-x+1\right)+3=3 .
\end{array}
$$