Three, (Full marks 20 points) Solve the equation:
$$
|x-| 2 x+1||=3 .
$$
Three, from the original equation, we get
$$
x^{2}+(2 x+1)^{2}-2 x|2 x+1|=9 \text {. }
$$

When $x \geqslant-\frac{1}{2}$, equation (1) becomes
$$
x^{2}+2 x-8=0 .
$$

Solving, we get $x_{1}=-4, x_{2}=2$.
$$
\because x \geqslant-\frac{1}{2} \text {, }
$$
$\therefore$ Therefore, $x_{2}=2$ is a root of the original equation.
When $x<-\frac{1}{2}$, equation (1) becomes
$$
9 x^{2}+6 x-8=0 \text {. }
$$

Solving, we get $x_{3}=\frac{2}{3}$ (which is not a solution since $x<-\frac{1}{2}$, so it is discarded),
$$
\dot{x}_{4}=-\frac{4}{3}
$$

Therefore, the original equation has two different solutions $x=-\frac{4}{3}, x=2$.