1. **State the problem:** Simplify the expression $$(6\sqrt{10} - 8\sqrt{2})(\sqrt{12} + 2\sqrt{10})$$. 2. **Recall the distributive property:** To multiply two binomials, use $$(a+b)(c+d) = ac + ad + bc + bd$$. 3. **Apply the distributive property:** $$6\sqrt{10} \times \sqrt{12} + 6\sqrt{10} \times 2\sqrt{10} - 8\sqrt{2} \times \sqrt{12} - 8\sqrt{2} \times 2\sqrt{10}$$ 4. **Simplify each term:** - $6\sqrt{10} \times \sqrt{12} = 6\sqrt{10 \times 12} = 6\sqrt{120}$ - $6\sqrt{10} \times 2\sqrt{10} = 12\sqrt{10 \times 10} = 12\sqrt{100}$ - $-8\sqrt{2} \times \sqrt{12} = -8\sqrt{2 \times 12} = -8\sqrt{24}$ - $-8\sqrt{2} \times 2\sqrt{10} = -16\sqrt{2 \times 10} = -16\sqrt{20}$ 5. **Simplify the radicals:** - $\sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30}$ - $\sqrt{100} = 10$ - $\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$ - $\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$ 6. **Substitute back:** $$6 \times 2\sqrt{30} + 12 \times 10 - 8 \times 2\sqrt{6} - 16 \times 2\sqrt{5}$$ $$= 12\sqrt{30} + 120 - 16\sqrt{6} - 32\sqrt{5}$$ 7. **Final simplified expression:** $$\boxed{12\sqrt{30} + 120 - 16\sqrt{6} - 32\sqrt{5}}$$