\[\boxed{\text{1089\ (1089).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]
\[Пусть\ \ a = \sqrt[3]{65 + x},\]
\[b = \sqrt[3]{65 - x} \Longrightarrow\]
\[a^{2} + 4b^{2} - 5ab = 0\]
\[\left( a^{2} - ab \right) + \left( 4b^{2} - 4ab \right) = 0\]
\[a(a - b) - 4b(a - b) = 0\]
\[(a - 4b)(a - b) = 0.\]
\[1)\ a = b,\]
\[\sqrt[3]{65 + x} = \sqrt[3]{65 - x} \Longrightarrow x = 0;\]
\[2)\ a = 4b,\]
\[\sqrt[3]{65 + x} = 4\sqrt[3]{65 - x} \Longrightarrow x = 63.\]
\[Ответ:\ x = 0;x = 63.\]