1. **State the problem:** Simplify the expression $5. (6\sqrt{3} - 5)(2\sqrt{5} + 4\sqrt{2})$. 2. **Use the distributive property (FOIL) to expand:** $$ (6\sqrt{3} - 5)(2\sqrt{5} + 4\sqrt{2}) = 6\sqrt{3} \times 2\sqrt{5} + 6\sqrt{3} \times 4\sqrt{2} - 5 \times 2\sqrt{5} - 5 \times 4\sqrt{2} $$ 3. **Calculate each term:** - $6\sqrt{3} \times 2\sqrt{5} = 12 \sqrt{15}$ - $6\sqrt{3} \times 4\sqrt{2} = 24 \sqrt{6}$ - $-5 \times 2\sqrt{5} = -10 \sqrt{5}$ - $-5 \times 4\sqrt{2} = -20 \sqrt{2}$ 4. **Combine the terms:** $$ 12 \sqrt{15} + 24 \sqrt{6} - 10 \sqrt{5} - 20 \sqrt{2} $$ 5. **Multiply the entire expression by 5:** $$ 5 \times (12 \sqrt{15} + 24 \sqrt{6} - 10 \sqrt{5} - 20 \sqrt{2}) = 60 \sqrt{15} + 120 \sqrt{6} - 50 \sqrt{5} - 100 \sqrt{2} $$ 6. **Final simplified expression:** $$ \boxed{60 \sqrt{15} + 120 \sqrt{6} - 50 \sqrt{5} - 100 \sqrt{2}} $$