1. **State the problem:** Simplify the expression $$(2\sqrt{3} - 5)(2\sqrt{3} + 5)$$. 2. **Formula used:** This is a product of conjugates, which follows the difference of squares formula: $$ (a - b)(a + b) = a^2 - b^2 $$ 3. **Identify terms:** Here, $a = 2\sqrt{3}$ and $b = 5$. 4. **Apply the formula:** $$ (2\sqrt{3})^2 - 5^2 $$ 5. **Calculate each square:** $$ (2\sqrt{3})^2 = 2^2 \times (\sqrt{3})^2 = 4 \times 3 = 12 $$ $$ 5^2 = 25 $$ 6. **Subtract:** $$ 12 - 25 = -13 $$ 7. **Final answer:** $$ -13 $$ The expression simplifies to $-13$.