1. Solve the inequality $\left(x^{2}+x+1\right)^{4 x^{3}+x^{2}} \geq\left(x^{2}+x+1\right)^{x-2 x^{3}}$.
Answer: $x \in\left[-1 ;-\frac{1}{2}\right] \cup\{0\} \cup\left[\frac{1}{3} ;+\infty\right)$.

Solution. The base of the exponent is positive for all $x$, so the given inequality is equivalent to the following:

$$
\left(x^{2}+x\right)\left(\left(4 x^{3}+x^{2}\right)-\left(x-2 x^{3}\right)\right) \geq 0 \Leftrightarrow(x+1) x^{2}\left(6 x^{2}+x-1\right) \geq 0 \Leftrightarrow(x+1) x^{2}(2 x+1)(3 x-1) \geq 0
$$

from which, using the method of intervals, we find that $x \in\left[-1 ;-\frac{1}{2}\right] \cup\{0\} \cup\left[\frac{1}{3} ;+\infty\right)$.