1. **State the problem:** Solve the equation $$\sqrt{2}x + 2 = -2 + \sqrt{6}x - 6$$ for $x$. 2. **Rewrite the equation:** Move all terms to one side to isolate the square root terms and constants: $$\sqrt{2}x + 2 + 2 - \sqrt{6}x + 6 = 0$$ which simplifies to $$\sqrt{2}x - \sqrt{6}x + 10 = 0$$ 3. **Combine like terms:** Factor $x$ out of the terms with square roots: $$x(\sqrt{2} - \sqrt{6}) + 10 = 0$$ 4. **Isolate $x$:** $$x(\sqrt{2} - \sqrt{6}) = -10$$ 5. **Divide both sides by $(\sqrt{2} - \sqrt{6})$:** $$x = \frac{-10}{\sqrt{2} - \sqrt{6}}$$ 6. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate $(\sqrt{2} + \sqrt{6})$: $$x = \frac{-10(\sqrt{2} + \sqrt{6})}{(\sqrt{2} - \sqrt{6})(\sqrt{2} + \sqrt{6})}$$ 7. **Simplify the denominator using difference of squares:** $$(\sqrt{2})^2 - (\sqrt{6})^2 = 2 - 6 = -4$$ 8. **Substitute back:** $$x = \frac{-10(\sqrt{2} + \sqrt{6})}{-4}$$ 9. **Simplify the negatives:** $$x = \frac{10(\sqrt{2} + \sqrt{6})}{4}$$ 10. **Reduce the fraction:** $$x = \frac{5(\sqrt{2} + \sqrt{6})}{2}$$ **Final answer:** $$x = \frac{5}{2}(\sqrt{2} + \sqrt{6})$$