Answer
$(-\infty,-4)U(1,\infty)$
Work Step by Step
Step 1. Graph $f(x)=-\frac{1}{2}(x+4)(x-1)^3$ as shown in the figure.
Step 2. The x-intercepts form three intervals $(-\infty,-4),(-4,1),(1,\infty)$
Step 3. For $f(x)\lt0$, we can identify the solution intervals as $(-\infty,-4)U(1,\infty)$