Answer
$$x ∈[2.15, +\infty)$$
Also see the image below.
Work Step by Step
$(x+1)^2\leq x^3$
$(x+1)^2$ is represented as the Blue graph on the image above.
$x^3$ is represented as the Red graph on the image above.
We need to find the region, where the Blue graph is below the Red graph. That is the region from $A$ to $+\infty$.
$x ∈[2.15, +\infty)$