$\dfrac{{\left( {3{x^2} - x} \right)\left( {3 - {x^2}} \right)}}{{4{x^2} + x - 3}} < 0 \\ \Leftrightarrow \dfrac{{x\left( {3x - 1} \right)\left( {3 - {x^2}} \right)}}{{\left( {4x - 3} \right)\left( {x + 1} \right)}} < 0$
$\begin{gathered} \ \hfill \\ \xrightarrow{{\, - \,\,\,\,\,\,\, - \sqrt 3 \,\,\,\,\,\, + \,\,\,\, - 1\,\,\,\,\,\,\,\, - \,\,\,\,\,\,\,\,0\,\,\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\,\dfrac{1}{3}\,\,\,\,\,\, - \,\,\,\,\,\,\dfrac{3}{4}\,\,\,\,\,\,\,\,\, + \,\,\,\,\,\,\,\sqrt 3 \,\,\,\,\,\,\,\,\,\, - }} \hfill \\ \Rightarrow x \in \left( { - \infty ; - \sqrt 3 } \right) \cup \left( { - 1;0} \right) \cup \left( {\dfrac{1}{3};\dfrac{3}{4}} \right) \cup \left( {\sqrt 3 ; + \infty } \right) \hfill \\ \end{gathered} $
