1. **State the problem:** Simplify the function $$f(x) = \frac{\sqrt{x-6} + \sqrt{2x-2}}{\sqrt{x-1} - \sqrt{x-5}}.$$\n\n2. **Identify domain restrictions:** The expressions under the square roots must be non-negative:\n- $x-6 \geq 0 \Rightarrow x \geq 6$\n- $2x-2 \geq 0 \Rightarrow x \geq 1$\n- $x-1 \geq 0 \Rightarrow x \geq 1$\n- $x-5 \geq 0 \Rightarrow x \geq 5$\n\nThe most restrictive domain is $x \geq 6$.\n\n3. **Simplify the denominator by rationalizing:** Multiply numerator and denominator by the conjugate of the denominator $\sqrt{x-1} + \sqrt{x-5}$ to eliminate the square roots in the denominator:\n$$f(x) = \frac{(\sqrt{x-6} + \sqrt{2x-2})(\sqrt{x-1} + \sqrt{x-5})}{(\sqrt{x-1} - \sqrt{x-5})(\sqrt{x-1} + \sqrt{x-5})}.$$\n\n4. **Simplify the denominator using difference of squares:**\n$$ (\sqrt{x-1})^2 - (\sqrt{x-5})^2 = (x-1) - (x-5) = 4.$$\n\n5. **Expand the numerator:**\n$$ (\sqrt{x-6})(\sqrt{x-1}) + (\sqrt{x-6})(\sqrt{x-5}) + (\sqrt{2x-2})(\sqrt{x-1}) + (\sqrt{2x-2})(\sqrt{x-5}).$$\n\n6. **Combine the terms under single square roots:**\n$$ \sqrt{(x-6)(x-1)} + \sqrt{(x-6)(x-5)} + \sqrt{(2x-2)(x-1)} + \sqrt{(2x-2)(x-5)}.$$\n\n7. **Calculate each product inside the roots:**\n- $(x-6)(x-1) = x^2 - 7x + 6$\n- $(x-6)(x-5) = x^2 - 11x + 30$\n- $(2x-2)(x-1) = 2x^2 - 4x + 2$\n- $(2x-2)(x-5) = 2x^2 - 12x + 10$\n\n8. **Rewrite the numerator:**\n$$ \sqrt{x^2 - 7x + 6} + \sqrt{x^2 - 11x + 30} + \sqrt{2x^2 - 4x + 2} + \sqrt{2x^2 - 12x + 10}.$$\n\n9. **Final simplified form:**\n$$f(x) = \frac{\sqrt{x^2 - 7x + 6} + \sqrt{x^2 - 11x + 30} + \sqrt{2x^2 - 4x + 2} + \sqrt{2x^2 - 12x + 10}}{4}, \quad x \geq 6.$$