1. **State the problem:** Simplify the expression $\sqrt{50} + 4\sqrt{18} - 6\sqrt{8}$.\n\n2. **Recall the rule:** $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ and simplify square roots by factoring out perfect squares.\n\n3. **Simplify each term:**\n- $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$\n- $4\sqrt{18} = 4 \times \sqrt{9 \times 2} = 4 \times \sqrt{9} \times \sqrt{2} = 4 \times 3 \sqrt{2} = 12\sqrt{2}$\n- $6\sqrt{8} = 6 \times \sqrt{4 \times 2} = 6 \times \sqrt{4} \times \sqrt{2} = 6 \times 2 \sqrt{2} = 12\sqrt{2}$\n\n4. **Rewrite the expression:**\n$$5\sqrt{2} + 12\sqrt{2} - 12\sqrt{2}$$\n\n5. **Combine like terms:**\n$$ (5 + 12 - 12)\sqrt{2} = 5\sqrt{2}$$\n\n**Final answer:** $5\sqrt{2}$