Answer
$[-4,-2] \cup [2, \infty)$
Work Step by Step
1. Express the inequality in the form $f(x)<0, f(x)>0, f(x)\leq 0$, or $f(x)\geq 0,$
where $f$ is a polynomial function.
$x^3+4x^2 \geq 4x+16$,
$x^3+4x^2-4x-16 \geq 0$,
$x^2(x+4)-4(x+4) \geq 0$,
$(x^2-4)(x+4) \geq 0$,
$(x-2)(x+2)(x+4) \geq 0$
$f(x)=(x-2)(x+2)(x+4)$
2. Solve the equation $f(x)=0$. The real solutions are the boundary points.
$(x-2)(x+2)(x+4)=0$
$x=2$ or $x=-2$ or $x=-4$
3. Make a table or diagram: use the test values to make a table or diagram of the sign of each factor in each interval.
4. Test each interval's sign of $f(x)$ with a test value,
$\begin{array}{llll}
Intervals: & a=test.v. & f(a),signs & f(a) \geq 0 ? \\
& &(a-2)(a+2)(a+4) & \\
(-\infty,-4) & -6 & (-)(-)(-) & F\\
(-4,-2) & -3 & (-)(-)(+) & T\\
(-2,2) & 0 & (-)(+)(+) & F\\
(2,\infty) & 4 & (+)(+)(+) & T
\end{array}$
5. Write the solution set, selecting the interval or intervals that satisfy the given inequality. If the inequality involves $\leq$ or $\geq$, include the boundary points.
Solution set: $[-4,-2] \cup [2, \infty)$
