1. **State the problem:** Simplify the expression $$(6\sqrt{3} - 5)(2\sqrt{5} + 4\sqrt{2})$$. 2. **Formula used:** Use the distributive property (FOIL method) for multiplying two binomials: $$(a - b)(c + d) = ac + ad - bc - bd$$. 3. **Apply the distributive property:** $$6\sqrt{3} \times 2\sqrt{5} + 6\sqrt{3} \times 4\sqrt{2} - 5 \times 2\sqrt{5} - 5 \times 4\sqrt{2}$$ 4. **Multiply the terms:** $$12\sqrt{15} + 24\sqrt{6} - 10\sqrt{5} - 20\sqrt{2}$$ 5. **Final simplified expression:** $$12\sqrt{15} + 24\sqrt{6} - 10\sqrt{5} - 20\sqrt{2}$$ No like terms to combine further.