jmc

The given inequality is equivalent to $$\frac{1}{2x+3}+2>0,$$or $$\frac{4x+7}{2x+3}>0.$$If $x<-\frac{7}{4},$ then $4x+7<0$ and $2x+3<0,$ so the inequality is satisfied.

If $-\frac{7}{4}<x<-\frac{3}{2},$ then $4x+7>0$ and $2x+3<0,$ so the inequality is not satisfied.

If $x>-\frac{3}{2},$ then $4x+7>0$ and $2x+3>0,$ so the inequality is satisfied. Thus, the solution is $$x\in \boxed{\left(-\infty ,-\frac{7}{4}\right)\cup \left(-\frac{3}{2},\infty \right)}.$$