6.021. $8 x^{4}+x^{3}+64 x+8=0$.

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

6.021. $8 x^{4}+x^{3}+64 x+8=0$.
Solution.

Domain: $x \in R$.

From the condition we have

$$
\begin{aligned}
& \left(8 x^{4}+x^{3}\right)+(64 x+8)=0 \Leftrightarrow x^{3}(8 x+1)+8(8 x+1)=0 \Leftrightarrow \\
& \Leftrightarrow(8 x+1)\left(x^{3}+8\right)=0 \Leftrightarrow(8 x+1)(x+2)\left(x^{2}-2 x+4\right)=0
\end{aligned}
$$

From this, $8 x+1=0, x_{1}=-\frac{1}{8}$, or $x+2=0, x_{2}=-2$, or

$$
x^{2}-2 x+4=0, x_{3,4} \in \varnothing(D<0)
$$

Answer: $\quad x_{1}=-\frac{1}{8}, x_{2}=-2$.