How to find all solutions of $\tan(x) = 2 + \tan(3x)$ without a calculator? Find all solutions of the equation $\tan(x) = 2 + \tan(3x)$ where $0<x<2\pi$.
By replacing $\tan(3x)$ with $\dfrac{\tan(2x) + \tan(x)}{1-\tan(2x)\tan(x)}$ 
I've gotten to $\tan^3 (x) - 3 \tan^2 (x) + \tan(x) + 1 =0$.
I am not sure how to proceed from there without the use of a calculator.

Let $\tan x=t$. Then, we have
$$t=2+\frac{3t-t^3}{1-3t^2}\Rightarrow (t-2)(1-3t^2)=3t-t^3$$$$\Rightarrow t^3-3t^2+t+1\Rightarrow (t-1)(t^2-2t-1)=0.$$
Hence, we have
$$\tan x=t=1,1\pm\sqrt 2\Rightarrow x=\frac{\pi}{4}+n\pi,\frac{3}{8}\pi+n\pi,-\frac{\pi}{8}+n\pi$$
where $n\in\mathbb Z$.
