\(x=\sqrt[3]{8 + 3\sqrt{21}} + \sqrt[3]{8 - 3\sqrt{21}}\\ x^3=\left(\sqrt[3]{8 + 3\sqrt{21}} + \sqrt[3]{8 - 3\sqrt{21}}\right)^2\\ =\sqrt[3]{8 + 3\sqrt{21}} ^3+\sqrt[3]{8 - 3\sqrt{21}}^3+3\sqrt[3]{8 + 3\sqrt{21}}.\sqrt[3]{8 - 3\sqrt{21}}\left(\sqrt[3]{8 + 3\sqrt{21}} + \sqrt[3]{8 - 3\sqrt{21}}\right)\\ =8 + 3\sqrt{21}+8 - 3\sqrt{21}+3\sqrt[3]{(8 + 3\sqrt{21})(8 - 3\sqrt{21})}x\\ =16+3\sqrt[3]{-125}x\\ =16-15x\\ \Rightarrow x^3=16-15x\\ \Leftrightarrow x^3+15x-16=0\\ \Leftrightarrow x^3-x^2+x^2-x+16x-16=0\\ \Leftrightarrow x^2(x-1)+x(x-1)+16(x-1)=0\\ \Leftrightarrow (x-1)(x^2+x+16)=0\\ \Leftrightarrow x=1.\)