6.012. $(x-1)\left(x^{2}-3\right)+(2 x-1)\left(x^{2}+2\right)=3$.
Solution.

Domain of definition: $x \in R$.

## We have

$$
\begin{aligned}
& x^{3}-x^{2}-3 x+3+2 x^{3}-x^{2}+4 x-2=3 \Leftrightarrow \\
& \Leftrightarrow 3 x^{3}-2 x^{2}+x-2=0 \Leftrightarrow 3 x^{3}-3 x^{2}+x^{2}-x+2 x-2=0 \Leftrightarrow \\
& \Leftrightarrow 3 x^{2}(x-1)+x(x-1)+2(x-1)=0 \Leftrightarrow(x-1)\left(3 x^{2}+x+2\right)=0 \\
& x-1=0, x_{1}=1 \text { or } 3 x^{2}+x+2=0, x_{2,3} \in \varnothing(D<0)
\end{aligned}
$$

Answer: $x=1$.