\(x=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\\ \Rightarrow x^3=\left(\sqrt[3]{9+4\sqrt{5}}\right)^3+\left(\sqrt[3]{9-4\sqrt{5}}\right)^3+3\sqrt[3]{9+4\sqrt{5}}\sqrt[3]{9-4\sqrt{5}}\left(\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\right)\)
\(\Rightarrow x^3=9+4\sqrt{5}+9-4\sqrt{5}+3.x\\ \Rightarrow x^3=18+3x\)
\(y=\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\)
\(y^3=\left(\sqrt[3]{3+2\sqrt{2}}\right)^3+\left(\sqrt[3]{3-2\sqrt{2}}\right)^3+3\sqrt[3]{3+2\sqrt{2}}\sqrt[3]{3-2\sqrt{2}}\left(\sqrt[3]{3+2\sqrt{2}}+\sqrt[3]{3-2\sqrt{2}}\right)\)
\(\Rightarrow y^3=3+2\sqrt{2}+3-2\sqrt{2}+3y\)
\(\Rightarrow y^3=6+3y\)
\(P=x^3+y^3-3\left(x+y\right)+1993\)
\(P=18+3x+6+3y-3\left(x+y\right)+1993\)
\(P=2017+3\left(x+y\right)-3\left(x+y\right)\)
\(P=2017\)
