You have provided a polynomial equation $f(x) = x^4 - 3x^3 - 5x^2 + 9x - 2$ and mentioned that one root of this equation is $2 + \sqrt{3}$. Given the rational root theorem, the conjugate $2 - \sqrt{3}$ should also be a root of the polynomial. I will verify this and proceed to find the remaining roots of the equation.

Let's first confirm that $2 + \sqrt{3}$ and $2 - \sqrt{3}$ are indeed roots of the polynomial by substituting these values into the polynomial. Then, I will factor the polynomial to find the other roots.