4. Solve the equation

$$
\log _{3}(x+2) \cdot \log _{3}(2 x+1) \cdot\left(3-\log _{3}\left(2 x^{2}+5 x+2\right)\right)=1
$$
Problem 4.

Answer: $x=1$.

Solution. For admissible values $x>-\frac{1}{2}$ we have: $\log _{3}(x+2)>0$.

If $\log _{3}(2 x+1)0$ and the left side of the equation is negative.

Thus, all three factors on the left side of the equation are positive, and the roots should be sought only among those $x$ for which

$$
\left\{x>0,2 x^{2}+5 x+20$ and the discriminant of this equation as a quadratic in $t$ is $D(p)=p(p-1)^{2}(p-4)$, then

$$
\{D(p) \geqslant 0, p>0\} \Longleftrightarrow p \in\{1\} \cup[4 ;+\infty)
$$

But for $p \geqslant 4: x \geqslant 40$, so $p=1$ and $t=1$, which is satisfied only if $x=1$.