$6.259 x^{3}-(3 a-1) x^{2}+\left(2 a^{2}-3 a\right) x+2 a^{2}=0$.

Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.

$6.259 x^{3}-(3 a-1) x^{2}+\left(2 a^{2}-3 a\right) x+2 a^{2}=0$.
Solution. The given equation is a special case of the model equation (1) if we set $v=-1, u=\frac{3 a}{2}, w^{2}=\frac{a^{2}}{4}$.

Answer: $x_{1}=-1, x_{2}=a, x_{3}=2 a$.

$6.260(x-1)^{5}+(x+3)^{5}=242(x+1)$.

Solution. Let $y=x-1, z=x+3$. Then

$$
\left\{\begin{array} { l } 
{ z - y = 4 , } \\
{ y ^ { 5 } + z ^ { 5 } = 1 2 1 ( y + z ) }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
z-y=4, \\
(y+z)\left(z^{4}-z^{3} y+z^{2} y^{2}-z y^{3}+y^{4}\right)=121(y+z)
\end{array}\right.\right.
$$

1) $\left\{\begin{array}{l}z-y=4, \\ z+y=0\end{array} \Leftrightarrow\left\{\begin{array}{l}z=2, \\ y=-2\end{array} \Leftrightarrow x=-1\right.\right.$.
2) $\left\{\begin{array}{l}z-y=4, \\ z^{3}(z-y)-y^{3}(z-y)+z^{2} y^{2}=121\end{array} \Leftrightarrow\left\{\begin{array}{l}z-y=4, \\ 4\left(z^{3}-y^{3}\right)=121-z^{2} y^{2}\end{array} \Leftrightarrow\right.\right.$

$\Leftrightarrow\left\{\begin{array}{l}z-y=4, \\ 16\left(z^{2}+y^{2}+z y\right)=121-z^{2} y^{2}\end{array} \Leftrightarrow\left\{\begin{array}{l}z-y=4, \\ 16(16+3 z y)=121-z^{2} y^{2}\end{array} \Leftrightarrow\right.\right.$

$\Leftrightarrow\left\{\begin{array}{l}z-y=4, \\ {\left[\begin{array}{l}z y=-3, \\ z y=-45\end{array}\right.}\end{array} \Leftrightarrow\left[\begin{array}{l}y=-1, z=3, \\ y=-3, z=1\end{array} \Leftrightarrow\left[\begin{array}{l}x=0, \\ x=-2 .\end{array}\right.\right.\right.$

Answer: $x_{1}=-1, x_{2}=0, x_{3}=-2$.