For the product to be $≤0$, one of the values $2x+3$ and $x-1$ has to be $≥0$ and the other has to be $≤0$. Consider the case when $2x+3\geq 0$ and $x-1\leq 0$.
$$2x+3\geq 0$$ $$x-1\leq 0$$
The solution satisfying both inequalities is $x\in \left[-\frac{3}{2},1\right]$.
