Example 3. When $x=\frac{1+\sqrt{1994}}{2}$, the value of the polynomial
$$
\left(4 x^{3}-1997 x-1994\right)^{2001}
$$

is ( ).
(A) 1
(B) -1
(C) $2^{2001}$
(D) $-2^{2001}$
Analyzing direct substitution, it is certainly not the best strategy. From the given, we have $2 x-1=\sqrt{19} 94$. Squaring and rearranging gives
$$
4 x^{2}-4 x-1993=0 \text {. }
$$

And
$$
\begin{array}{l}
4 x^{3}-1997 x-1994 \\
=(x-1)\left(4 x^{2}-4 x-1993\right)-1 \\
=-1 .
\end{array}
$$

Therefore, the desired result is $(-1)^{2001}=-1$.