1. **Problem statement:** Simplify the expressions: (i) $ (4\sqrt{3} - 2\sqrt{2})(3\sqrt{2} + 4\sqrt{3}) $ (ii) $ (2 + \sqrt{3})(3 + \sqrt{5}) $ (iii) $ (\sqrt{3} + \sqrt{2})^2 $ 2. **Formula and rules:** - Use distributive property (FOIL) for multiplication: $ (a+b)(c+d) = ac + ad + bc + bd $ - Use $ (a+b)^2 = a^2 + 2ab + b^2 $ - Simplify square roots and combine like terms. --- 3. **Step-by-step simplification:** (i) $ (4\sqrt{3} - 2\sqrt{2})(3\sqrt{2} + 4\sqrt{3}) $ = $ 4\sqrt{3} \times 3\sqrt{2} + 4\sqrt{3} \times 4\sqrt{3} - 2\sqrt{2} \times 3\sqrt{2} - 2\sqrt{2} \times 4\sqrt{3} $ = $ 12\sqrt{6} + 16 \times 3 - 6 \times 2 - 8\sqrt{6} $ = $ 12\sqrt{6} + 48 - 12 - 8\sqrt{6} $ Combine like terms: = $ (12\sqrt{6} - 8\sqrt{6}) + (48 - 12) $ = $ 4\sqrt{6} + 36 $ --- (ii) $ (2 + \sqrt{3})(3 + \sqrt{5}) $ = $ 2 \times 3 + 2 \times \sqrt{5} + \sqrt{3} \times 3 + \sqrt{3} \times \sqrt{5} $ = $ 6 + 2\sqrt{5} + 3\sqrt{3} + \sqrt{15} $ No like terms to combine, so the simplified form is: = $ 6 + 2\sqrt{5} + 3\sqrt{3} + \sqrt{15} $ --- (iii) $ (\sqrt{3} + \sqrt{2})^2 $ = $ (\sqrt{3})^2 + 2 \times \sqrt{3} \times \sqrt{2} + (\sqrt{2})^2 $ = $ 3 + 2\sqrt{6} + 2 $ = $ 5 + 2\sqrt{6} $ --- **Final answers:** (i) $ 4\sqrt{6} + 36 $ (ii) $ 6 + 2\sqrt{5} + 3\sqrt{3} + \sqrt{15} $ (iii) $ 5 + 2\sqrt{6} $