Let $r$ be a root of the polynomial $p(x) = (\sqrt{5} - 2\sqrt{3})x^3 + \sqrt{3}x - \sqrt{5} + 1$. Find another polynomial $q(x)$ with integer coefficients such that $q(r) = 0$. 
I have no clue how to do this question. Can't use rational root theorem and I see no feasible way to get the roots of $p(x)$. Any help would be appreciated. 

It's $$5(x^3-1)^2=(\sqrt3(2x^3-x)-1)^2$$ and from here
$$(5(x^3-1)^2-3(2x^3-x)^2-1)^2=12(2x^3-x)^2,$$
which is
$$(7x^6-12x^4+10x^3+3x^2-4)^2-12x^2(2x^2-1)^2=0$$ or
$$49x^{12}-168x^{10}+140x^9+186x^8-240x^7-76x^6+60x^5+153x^4-80x^3-36x^2+16=0.$$
