1. **State the problem:** Expand and fully simplify the expression $$(7 + \sqrt{3})(1 + \sqrt{3})$$. 2. **Formula used:** Use the distributive property (FOIL method) for multiplying binomials: $$(a + b)(c + d) = ac + ad + bc + bd$$. 3. **Apply the formula:** $$7 \times 1 + 7 \times \sqrt{3} + \sqrt{3} \times 1 + \sqrt{3} \times \sqrt{3}$$ 4. **Calculate each term:** $$7 + 7\sqrt{3} + \sqrt{3} + 3$$ 5. **Combine like terms:** $$7 + 3 + (7\sqrt{3} + \sqrt{3}) = 10 + 8\sqrt{3}$$ 6. **Final answer:** $$10 + 8\sqrt{3}$$