$\begin{array}{l}
\sqrt[3]{{7 + 5\sqrt 2 }} = \sqrt[3]{{2\sqrt 2 + 3.{{\sqrt 2 }^2}.1 + 1 + 3.1\sqrt 2 }}\\
= \sqrt[3]{{{{\left( {\sqrt 2 + 1} \right)}^3}}} = \sqrt 2 + 1\\
\dfrac{1}{{\sqrt[3]{{7 + 5\sqrt 2 }}}} = \dfrac{1}{{\sqrt[3]{{{{\left( {\sqrt 2 + 1} \right)}^3}}}}} = \dfrac{1}{{\sqrt 2 + 1}} = \sqrt 2 - 1\\
\Rightarrow x = \sqrt[3]{{7 + 5\sqrt 2 }} - \dfrac{1}{{\sqrt[3]{{7 + 5\sqrt 2 }}}} = 2\\
\Rightarrow {x^3} + 3x - 14 = {2^3} + 3.2 - 14 = 0
\end{array}$
