1. **State the problem:** Multiply and simplify the expression $$(5 - 9\sqrt{3})(4 - 8\sqrt{3})$$. 2. **Formula used:** Use the distributive property (FOIL method) to multiply two binomials: $$(a - b)(c - d) = ac - ad - bc + bd$$. 3. **Apply the formula:** $$5 \times 4 = 20$$ $$5 \times (-8\sqrt{3}) = -40\sqrt{3}$$ $$-9\sqrt{3} \times 4 = -36\sqrt{3}$$ $$-9\sqrt{3} \times (-8\sqrt{3}) = 72 \times 3 = 216$$ 4. **Combine all terms:** $$20 - 40\sqrt{3} - 36\sqrt{3} + 216$$ 5. **Simplify like terms:** $$20 + 216 - (40\sqrt{3} + 36\sqrt{3}) = 236 - 76\sqrt{3}$$ **Final answer:** $$236 - 76\sqrt{3}$$