4. Find all solutions to the equation

$$
8^{x}(3 x+1)=4
$$

and prove that there are no other solutions.
Solution. It is easy to notice that $x=\frac{1}{3}$ is a solution. For $x<-\frac{1}{3}$, the left side of the equation is negative, while the right side is positive, so there can be no roots. For $x \geq-\frac{1}{3}$, the left side monotonically increases, therefore, the equation can have only one root, so there are no other roots.