Lake center. Factorize in the set of real numbers $k$:
$$
3 x^{3}-4 x+1
$$
\begin{aligned} \text { Original expression }= & \left(3 x^{3}-3 x\right)-(x-1) \\ = & (x-1)\left(3 x^{2}+3 x-1\right) \\ = & 3(x-1)\left(x+\frac{3-\sqrt{21}}{6}\right) \\ & \cdot\left(x+\frac{3+\sqrt{21}}{6}\right)\end{aligned}