#### Answer

$(-1, -0.25) \cup(-0.25, +\infty)$

#### Work Step by Step

Add $9x+1$ to both sides of the inequality to obtain:
$16x^3+24x^2+9x+1\gt -9x-1+9x+1
\\16x^3+24x^2+9x+1\gt 0
$
Graph the function $y=16x^3+24x^2+9x+1$
(refer to the attached image below for the graph)
The solution to the given inequality is the set of x-values for which $y\gt 0$.
Notice that the graph shows $y \gt 0$ in the following intervals:
$(-1, -0.25)$ and $(-0.25, +\infty)$