Try to solve this equation without multiplying it out. Notice a symmetry and make a substitution to a new variable and you can quickly solve this quartic by turning it into a quadratic...
 

If you are trying to find the square root of $4+2\sqrt{3}$ (i.e. you are trying to find $x$ such that $x^2=4+2\sqrt{3}$) then you can write $x=a+b\sqrt{3}$ and then find $a$ and $b$ such that $(a+b\sqrt{3})^2=4+2\sqrt{3}$.