1. **State the problem:** Multiply and simplify the expression $$(5x - 3\sqrt{3})(5x + 3\sqrt{3})$$. 2. **Recall the formula:** This is a product of conjugates, which follows the difference of squares formula: $$ (a - b)(a + b) = a^2 - b^2 $$ where $a = 5x$ and $b = 3\sqrt{3}$. 3. **Apply the formula:** $$ (5x)^2 - (3\sqrt{3})^2 $$ 4. **Calculate each square:** $$ (5x)^2 = 25x^2 $$ $$ (3\sqrt{3})^2 = 3^2 \times (\sqrt{3})^2 = 9 \times 3 = 27 $$ 5. **Substitute back:** $$ 25x^2 - 27 $$ 6. **Final simplified expression:** $$ \boxed{25x^2 - 27} $$ This is the simplified product of the given binomials.