1. **State the problem:** Simplify the expression $2\sqrt{3}(2\sqrt{6} + 5)$.\n\n2. **Recall the distributive property:** $a(b + c) = ab + ac$. We will distribute $2\sqrt{3}$ to both terms inside the parentheses.\n\n3. **Apply the distributive property:**\n$$2\sqrt{3} \times 2\sqrt{6} + 2\sqrt{3} \times 5$$\n\n4. **Multiply the coefficients and the radicals separately:**\n$$2 \times 2 \times \sqrt{3} \times \sqrt{6} + 2 \times 5 \times \sqrt{3}$$\n$$= 4 \times \sqrt{3 \times 6} + 10\sqrt{3}$$\n\n5. **Simplify inside the square root:**\n$$\sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}$$\n\n6. **Substitute back:**\n$$4 \times 3\sqrt{2} + 10\sqrt{3} = 12\sqrt{2} + 10\sqrt{3}$$\n\n7. **Final answer:**\n$$\boxed{12\sqrt{2} + 10\sqrt{3}}$$