What is the domain of the function $g(x) = x^3 + \sqrt{x + 2} - 10$?

The cube and the constant term are defined for every real $x$. The square root $\sqrt{x + 2}$ requires its argument to be nonnegative: $x + 2 \geq 0$, that is $x \geq -2$. So the domain consists of all real numbers at least $-2$. At $x = -2$ the root equals 0, which is allowed.