1. The problem is to simplify the expression where the first square root covers $(2x+2)$ and the second square root covers $(6x-6)$. 2. The expression is $$\sqrt{2x+2} - \sqrt{6x-6}$$. 3. First, factor each radicand: $$2x+2 = 2(x+1)$$ $$6x-6 = 6(x-1)$$ 4. Rewrite the expression using these factorizations: $$\sqrt{2(x+1)} - \sqrt{6(x-1)}$$ 5. Use the property of square roots: $$\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$$ 6. So, $$\sqrt{2} \cdot \sqrt{x+1} - \sqrt{6} \cdot \sqrt{x-1}$$ 7. Since $\sqrt{6} = \sqrt{2 \cdot 3} = \sqrt{2} \cdot \sqrt{3}$, rewrite: $$\sqrt{2} \cdot \sqrt{x+1} - \sqrt{2} \cdot \sqrt{3} \cdot \sqrt{x-1}$$ 8. Factor out $\sqrt{2}$: $$\sqrt{2} \left( \sqrt{x+1} - \sqrt{3} \cdot \sqrt{x-1} \right)$$ 9. This is the simplified form of the original expression. Final answer: $$\boxed{\sqrt{2} \left( \sqrt{x+1} - \sqrt{3} \sqrt{x-1} \right)}$$