Answer
$[-4,-1]U[1,\infty)$. See graph.
Work Step by Step
Step 1. $x^3+4x^2\ge x+4\longrightarrow x^2(x+4)\ge x+4\longrightarrow (x+4)(x^2-1)\ge0 \longrightarrow (x+4)(x+1)(x-1)\ge0$
Step 2. Identify boundary points $x=-4,-1,1$ and intervals $(-\infty,-4],[-4,-1],[-1,1],[1,\infty)$
Step 3. Choose test values $x=-5,-2,0,2$ for each interval and evaluate the inequality to get $False,\ True,\ False,\ True$
Step 4. Thus we have the solution interval(s) $[-4,-1]U[1,\infty)$. See graph.