1. **State the problem:** We want to find a common denominator for the expression $$\frac{2x}{6x+6} + \frac{x-8}{12x-12}$$. 2. **Factor each denominator:** - Factor the first denominator: $$6x+6 = 6(x+1)$$. - Factor the second denominator: $$12x-12 = 12(x-1)$$. 3. **Find the least common denominator (LCD):** - The denominators are $$6(x+1)$$ and $$12(x-1)$$. - Factor constants: 6 and 12. The least common multiple of 6 and 12 is 12. - The LCD must include both factors $$x+1$$ and $$x-1$$. - Therefore, the LCD is $$12(x+1)(x-1)$$. 4. **Rewrite each fraction with the LCD:** - For $$\frac{2x}{6(x+1)}$$, multiply numerator and denominator by $$2(x-1)$$ to get denominator $$12(x+1)(x-1)$$. - For $$\frac{x-8}{12(x-1)}$$, multiply numerator and denominator by $$x+1$$ to get denominator $$12(x+1)(x-1)$$. 5. **Final expression with common denominator:** $$\frac{2x \cdot 2(x-1)}{12(x+1)(x-1)} + \frac{(x-8)(x+1)}{12(x+1)(x-1)}$$ This is how you get a common denominator for the given expression.