[tex]\sqrt[3]x +2\sqrt[3]{x^2} = 3 \\ \\ \sqrt[3]x +2(\sqrt[3]{x})^2 = 3 \\ \\ \\\sqrt[3]{x} = t,\quad t\in \mathbb{R} \\ \\\\ t +2t^2 = 3 \\ \\ 2t^2+t-3 = 0\\ \\ \Delta = 1-4\cdot 2\cdot(-3) = 1+24 = 25 \Rightarrow t_{1,2} = \dfrac{-1\pm 5}{4}[/tex]
[tex]\Rightarrow \left|\begin{array}{lcl}t=1\\ t=-\dfrac{3}{2}\end{array} \Rightarrow \left|\begin{array}{lcl}\sqrt[3]{x} =1\\ \sqrt[3]{x}=-\dfrac{3}{2}\end{array} \Rightarrow \left|\begin{array}{lcl}x =1\\ x=-{\dfrac{27}{8}}\end{array} \Rightarrow x\in \Big\{-{\dfrac{27}{8};1\Big\}[/tex]
[tex]\Rightarrow S = -\dfrac{27}{8}+1 = \dfrac{-27+8}{8} = \boxed{-\dfrac{19}{8}}[/tex]
