1. **State the problem:** Expand and simplify the expression $$(3 + \sqrt{7})(4 + \sqrt{7})$$. 2. **Formula used:** Use the distributive property (FOIL method) for binomials: $$(a + b)(c + d) = ac + ad + bc + bd$$. 3. **Apply the formula:** $$ (3 + \sqrt{7})(4 + \sqrt{7}) = 3 \times 4 + 3 \times \sqrt{7} + \sqrt{7} \times 4 + \sqrt{7} \times \sqrt{7} $$ 4. **Calculate each term:** $$ = 12 + 3\sqrt{7} + 4\sqrt{7} + 7 $$ 5. **Combine like terms:** $$ = (12 + 7) + (3\sqrt{7} + 4\sqrt{7}) = 19 + 7\sqrt{7} $$ 6. **Final answer:** $$ \boxed{19 + 7\sqrt{7}} $$