Denote $x^2 + x = a$ and $x^3 + 2x^2 = b$. Then $b - a x = x^2 = a - x$, hence $x(a - 1) = b - a$. Since $x$ is an irrational number and $a, b$ are integers, we deduce that $a = b = 1$, and, finally, $x = \frac{-1 \pm \sqrt{5}}{2}$.