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)=ax⁴+b√(3)x³+cx²+d√(3)x+e

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

Yes, I wanted you to see the picture.  It was my starting point for the problem.  Not at all a spoiler.