[tex](2 \sqrt{2} - 5 \sqrt{3})^{2} = \\ \\ {(2 \sqrt{2}) }^{2} - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + {(5 \sqrt{3}) }^{2} = \\ \\ {2}^{2} { \sqrt{2} }^{2} - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + {(5 \sqrt{3}) }^{2} = \\ \\ 4 { \sqrt{2} }^{2} - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + {(5 \sqrt{3}) }^{2} = \\ \\ 4 \times 2 - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + {(5 \sqrt{3}) }^{2} = \\ \\ 4 \times 2 - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + {5}^{2} { \sqrt{3} }^{2} = \\ \\ 4 \times 2 - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + 25 { \sqrt{3} }^{2} = \\ \\ 4 \times 2 - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + 25 \times 3 = \\ \\ 8 - 2 \times 2 \sqrt{2} \times 5 \sqrt{3} + 25 \times 3 = \\ \\ 8 - 2 \times 2 \times 5 \sqrt{2 \times 3} + 25 \times 3 = \\ \\ 8 - 2 \times 2 \times 5 \sqrt{6} + 25 \times 3 = \\ \\ 8 - 20 \sqrt{6} + 25 \times 3 = \\ \\ 8 - 20 \sqrt{6} + 75 = \\ \\ 83 - 20 \sqrt{6} [/tex]
