Cómo resolver la desigualdad $\frac{12}{2x-3}<1+2x$

Question

Intento resolver la siguiente desigualdad:

$$\frac{12}{2x-3}<1+2x$$

En la respuesta dada,

$$\frac{12}{2x-3}-(1+2x)<0$$

$$\frac{-(2x+3)(2x-5)}{2x-3}<0 \rightarrow \textrm{ How do I get to this step?}$$

$$\frac{(2x+3)(2x-5)}{2x-3}>0$$

$$(2x+3)(2x-5)(2x-3)>0 \textrm{ via multiply both sides by }(2x-3)^2$$

Answer 1

$$ \frac{12}{2x-3} - (1-2x) = \frac{12 - (1+2x)(2x-3) }{2x-3} = \frac{ 12 - (2x-3+4x^2-6x)}{2x-3} $$

$$= - \frac{4x^2-4x-15}{2x-3} = - \frac{(2x+3)(2x-5)}{2x-3} $$