Let's determine the value of the expression $x^{4}-8 x^{3}+10 x^{2}-8 x+1$ when $x$ is replaced by any root of the equation

$$
x^{2}-4 x+1=0
$$
We divide the given quartic expression by the quadratic $x^{2}-4 x+1$:

$$
f(x) \equiv x^{4}-8 x^{3}+10 x^{2}-8 x+1=\left(x^{2}-4 x+1\right)\left(x^{2}-4 x+7\right)-8(4 x-1) .
$$

For those values of $x$ for which $x^{2}-4 x+1=0$, the value of $f(x)$ coincides with the value of the linear expression $-8(4 x-1)$. The roots of $x^{2}-4 x+1=0$ are $2+\sqrt{3}$ and $2-\sqrt{3}$. Accordingly,

$$
\begin{aligned}
& f(2+\sqrt{3})=-8(8+4 \sqrt{3}-1)=-32 \sqrt{3}-56 \\
& f(2-\sqrt{3})=-8(8-4 \sqrt{3}-1)=32 \sqrt{3}-56
\end{aligned}
$$

Kovács Ibolya (Izr. leányg. VI. o. Bp.)