Can someone help me find the exact solutions to the equation

$\displaystyle x^3 + 2x^2 - 9x + 3 = 0$?

I've used a grapher to find approximate solutions at the three places where it crosses the x-axis, and tried my best to work through

Cardano's solution ($\displaystyle t^3 -\frac{31}{3}t + \frac{259}{27} = 0$) but I just can't finish it.

A big part of my problem here is that using the rational root theorem leads me to try $\displaystyle x = \pm 1, \pm 3$, none of which work, so all three real solutions (it crosses three times, so no complex-imaginary roots) must be irrational. I recognize it obviously must be possible, but I'm not used to three irrational roots giving such a nice, simplistic equation. These three irrationals must multiply to be perfectly 3, add to be perfectly -2, and something else (what?) to be perfectly -9 (or +9? I can't remember).

Obviously I've put a good amount of thought into this problem; somebody please help me out!