Answer
$$x∈(-1, -0.25)∪(0.25, +\infty)$$
Also see the image below.
Work Step by Step
$16x^3+24x^2>-9x-1$
$16x^3+24x^2$ is represented as the Blue graph on the image above.
$-9x-1$ is represented as the Red graph on the image above.
We have two ($A$ and $B$) intersections. We need the regions where the Red graph lies below (is less than) the Blue graph. That is from point $A$ to $+\infty$. But, note here, we have discontinuity at the point $B$, so actual solution will be in the region:
$(-1, -0.25)∪(0.25, +\infty)$
excluding point $B$.
