How to solve $2x^5+5\sqrt{2}x^4+20x^3+20\sqrt{2}x^2+20x+4\sqrt{2}=0$? 
How to solve $$2x^5+5\sqrt{2}x^4+20x^3+20\sqrt{2}x^2+20x+4\sqrt{2}=0?$$

I just have no idea and I'have some knowledge about the polynomial equations. Here, just nothing. 

Substitute $x = \sqrt{2}y$ and divide by $\sqrt{2}^5$. The resulting equation is
$$y^5 + (y+1)^5 = 0.$$
Now substitute $y = -1/(z+1)$ and multiply by $(z+1)^5$ and you get
$$z^5 - 1 = 0.$$
