1. **State the problem:** Multiply and simplify the expression $2\sqrt{6} ( 5\sqrt{6} - 4\sqrt{2} )$. 2. **Recall the distributive property:** $a(b - c) = ab - ac$. 3. **Apply the distributive property:** $$2\sqrt{6} \times 5\sqrt{6} - 2\sqrt{6} \times 4\sqrt{2}$$ 4. **Multiply coefficients and radicals separately:** $$= (2 \times 5)(\sqrt{6} \times \sqrt{6}) - (2 \times 4)(\sqrt{6} \times \sqrt{2})$$ 5. **Simplify the radicals:** $$\sqrt{6} \times \sqrt{6} = \sqrt{36} = 6$$ $$\sqrt{6} \times \sqrt{2} = \sqrt{12}$$ 6. **Substitute back:** $$= 10 \times 6 - 8 \times \sqrt{12}$$ 7. **Calculate the first term:** $$= 60 - 8\sqrt{12}$$ 8. **Simplify $\sqrt{12}$:** $$\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}$$ 9. **Substitute and simplify:** $$60 - 8 \times 2\sqrt{3} = 60 - 16\sqrt{3}$$ **Final answer:** $60 - 16\sqrt{3}$