Pack five unit circles in the smallest regular hexagon possible.
The exact solution for the smallest possible side length of the hexagon is given by the largest real root of a fourth degree polynomial:

P(x)=ax^{4}+b√(3)x^{3}+cx^{2}+d√(3)x+e

Where a,b,c,d,e are integers. Find them.

On the internet I found a drawing of the figure. The side length (root) is given as well, but not the polynomial nor its derivation. For me, this is a good place to start, but one might also fairly regard this as

**a spoiler:**

*Edited on ***November 25, 2018, 11:01 am**