Junior Macedonian Mathematical Olympiad

Let $a=x^3$, $b=x^2+x$. Then $a=x^3=x(x^2+x)-(x^2+x)=xb-b=b(x-1)$. It is clear that $b\ne -1$, since, if this wasn't the case, $x^2+x=-1$ or $(x+\frac{1}{2}{)}^2-\frac{1}{4}=-1$. Then $(x+\frac{1}{2}{)}^2=-\frac{3}{4}$ which is impossible. We get that $x=\frac{a+b}{b+1}$ which is a rational number since the numbers $a$ and $b$ are rational.