$6.27010 x^{3}-3 x^{2}-2 x+1=0$.
## Solution.

$$
10 x^{3}+5 x^{2}-8 x^{2}-4 x+2 x+1=0 \Leftrightarrow 5 x^{2}(2 x+1)-4 x(2 x+1)+(1+2 x)=0 \Leftrightarrow
$$

$\Leftrightarrow(2 x+1)\left(5 x^{2}-4 x+1\right)=0 \Leftrightarrow 2 x+1=0$.

Answer: $x=-\frac{1}{2}$.

$6.2712\left(x^{2}+x+1\right)^{2}-7(x-1)^{2}=13\left(x^{3}-1\right)$.

Solution. Let $v=x^{2}+x+1, w=x-1$, then $2 v^{2}-7 w^{2}=13 v w \Leftrightarrow$

$$
\Leftrightarrow 2 v^{2}-13 v w-7 w^{2}=0 \Leftrightarrow v=\frac{13 w \pm \sqrt{169 w^{2}+56 w^{2}}}{4}=\frac{(13 \pm 15) w}{4} \Leftrightarrow
$$

$$
\Leftrightarrow\left[\begin{array} { c } 
{ v = 7 w , } \\
{ v = - \frac { 1 } { 2 } w , }
\end{array} \Leftrightarrow \left[\begin{array} { l } 
{ x ^ { 2 } - 6 x + 8 = 0 , } \\
{ 2 x ^ { 2 } + 3 x + 1 = 0 , }
\end{array} \Leftrightarrow \left[\begin{array}{c}
(x-2)(x-4)=0 \\
(x+1)\left(x+\frac{1}{2}\right)=0,
\end{array} \Leftrightarrow\right.\right.\right.
$$

Answer: $x_{1}=2, x_{2}=4, x_{3}=-1, x_{4}=-\frac{1}{2}$.

## $6.27227 x^{3}+9 x^{2}-48 x+20=0$.

Solution. Let $y=3 x$, then $y^{3}+y^{2}-16 y+20=0, y=2$ is a root of this equation $\Rightarrow(y-2)\left(y^{2}+3 y-10\right)=0 \Leftrightarrow(y-2)(y-2)(y+5)=0 \Leftrightarrow$ $\Leftrightarrow\left[\begin{array}{l}y=2, \\ y=2, \\ y=-5\end{array} \Leftrightarrow\left[\begin{array}{l}3 x=2, \\ 3 x=2, \\ 3 x=-5 .\end{array}\right.\right.$

Answer: $x_{1}=x_{2}=\frac{2}{3}, x_{3}=-\frac{5}{3}$.