How to solve $x+x^2+x^3+x^4+x^5=62$? $$x+x^2+x^3+x^4+x^5=62$$
The roots in the set of Real numbers is $x=2$, but how solve it?

Apply the rational roots theorem:
$$62\implies1,2,31,62$$
Thus, it is easy enough to check if $x=\pm1,\pm2,\pm31,\pm62$ are roots.  Indeed, if we check, we find that $x=2$ is the only rational root.  The remaining roots can be deduced to be complex since after factoring out, we find it has only one real relative minimum at $x=-1.55$ such that $\left(x^4+3 x^3+7 x^2+15 x+31\right)>0$ for all $x$.
