Simplify: $\;$ $\dfrac{2x^{\frac{-1}{3}}}{x^{\frac{2}{3}} - 3 x^{\frac{-1}{3}}} - \dfrac{x^{\frac{2}{3}}}{x^{\frac{5}{3}} - x^{\frac{2}{3}}} - \dfrac{x + 1}{x^2 - 4x + 3}$
$\dfrac{2x^{\frac{-1}{3}}}{x^{\frac{2}{3}} - 3 x^{\frac{-1}{3}}} - \dfrac{x^{\frac{2}{3}}}{x^{\frac{5}{3}} - x^{\frac{2}{3}}} - \dfrac{x + 1}{x^2 - 4x + 3}$
$= \dfrac{\dfrac{2}{x^{\frac{1}{3}}}}{x^{\frac{2}{3}} - \dfrac{3}{x^{\frac{1}{3}}}} - \dfrac{x^{\frac{2}{3}}}{x^{\frac{2}{3}} \left(x - 1\right)} - \dfrac{x + 1}{x^2 - 3x - x + 3}$
$= \dfrac{2}{x - 3} - \dfrac{1}{x - 1} - \dfrac{x + 1}{\left(x - 3\right) \left(x - 1\right)}$
$= \dfrac{2 x - 2 - x + 3 - x - 1}{\left(x - 3\right) \left(x - 1\right)}$
$= \dfrac{2x -2x + 3 - 3}{\left(x - 3\right) \left(x - 1\right)}$
$= 0$