Answer
$ (-1,\frac{2}{3})\cup(2,\infty)$.
Work Step by Step
Step 1. For $\frac{(2-x)^3(3x-2)}{(x+1)(x^2-x+1)}\lt0$, identify boundary points as $x=-1,\frac{2}{3},2$.
Step 2. Form intervals $(-\infty,-1),(-1,\frac{2}{3}),(\frac{2}{3},2),(2,\infty)$.
Step 3. Choose test values for each interval $x=-2,0,1,3$.
Step 4. Test the inequality to get results: $False,\ True,\ False,\ True$
Step 5. We have the solution $ (-1,\frac{2}{3})\cup(2,\infty)$.
