Answer
$(-\infty, -1)\cup(0,\frac{5}{2})$.
Work Step by Step
Step 1. Factor the inequality to get $x(2x^2-3x-5)\lt0\longrightarrow x(2x-5)(x+1)\lt0$
Step 2. Identify the boundary points $x=-1, 0, \frac{5}{2}$ and separate the number line into intervals $(-\infty, -1)$, $(-1,0)$, $(0,\frac{5}{2})$ and $(\frac{5}{2},\infty)$
Step 3. Use test points $x=-2,-\frac{1}{2},1,3$ to get the signs of the left side of the inequality as $-,+,-,+$
Step 4. Based on the signs, we have the solution as $(-\infty, -1)\cup(0,\frac{5}{2})$.
