$\begin{array}{l} x = \sqrt[3]{{4\left( {\sqrt 5 + 1} \right)}} - \sqrt[3]{{4\left( {\sqrt 5 - 1} \right)}}\\ \Leftrightarrow {x^3} = 4\sqrt 5 + 4 - 4\sqrt 5 + 4 - 3\sqrt[3]{{16\left( {\sqrt 5 - 1} \right)\left( {\sqrt 5 + 1} \right)}}.\left[ {\sqrt[3]{{4\left( {\sqrt 5 + 1} \right)}} - \sqrt[3]{{4\left( {\sqrt 5 - 1} \right)}}} \right]\\ \Leftrightarrow {x^3} = 8 - 3.\sqrt[3]{{16.4}}.x\\ \Leftrightarrow {x^3} = 8 - 12x\\ \Leftrightarrow {x^3} - 12x + 8 = 0\\ \end{array}$
