C1: $x=\sqrt[3]{(\sqrt 2+1)^3}-\dfrac1{\sqrt[3]{(\sqrt 2+1)^3}}=\sqrt 2+1-\dfrac1{\sqrt 2+1}=\sqrt 2+1-(\sqrt 2-1)=2\Rightarrow x^3+3x-14=0$
C2: $x^3=7+5\sqrt 2 - \dfrac1{7+5\sqrt 2}-3\sqrt[3]{7+5\sqrt 2}.\dfrac1{\sqrt[3]{7+5\sqrt 2}}.(\sqrt[3]{7+5\sqrt 2}-\dfrac1{\sqrt[3]{7+5\sqrt 2}})$
$\Leftrightarrow x^3=7+5\sqrt 2-\dfrac{7-5\sqrt 2}{(7+5\sqrt 2)(7-5\sqrt 2)}-3x$
$\Leftrightarrow x^3=7+5\sqrt 2-(5\sqrt 2-7)-3x$
$\Leftrightarrow x^3=14-3x\Leftrightarrow x^3+3x-14=0$