1. **State the problem:** Multiply and simplify the expression $$(6 + \sqrt{2})(7 + 8\sqrt{2})$$. 2. **Recall the distributive property (FOIL method):** To multiply two binomials, multiply each term in the first binomial by each term in the second binomial. 3. **Apply the distributive property:** $$6 \times 7 = 42$$ $$6 \times 8\sqrt{2} = 48\sqrt{2}$$ $$\sqrt{2} \times 7 = 7\sqrt{2}$$ $$\sqrt{2} \times 8\sqrt{2} = 8 \times 2 = 16$$ (since $\sqrt{2} \times \sqrt{2} = 2$) 4. **Combine all terms:** $$42 + 48\sqrt{2} + 7\sqrt{2} + 16$$ 5. **Simplify like terms:** $$42 + 16 + (48\sqrt{2} + 7\sqrt{2}) = 58 + 55\sqrt{2}$$ **Final answer:** $$58 + 55\sqrt{2}$$