1. **State the problem:** We are given two expressions: (2) $$\frac{1}{a+1} - \frac{1}{a+3} - \frac{1}{a+2} + \frac{1}{a+4}$$ (3) $$\frac{1}{a-1} - \frac{1}{a+1} - \frac{2}{a^2+1} - \frac{4}{a^4+1}$$ The task is to first find the common denominator of the terms $$\frac{1}{a-1}$$ and $$\frac{1}{a+1}$$ in expression (3). 2. **Formula and rules:** To add or subtract fractions, we need a common denominator. For two fractions $$\frac{1}{x}$$ and $$\frac{1}{y}$$, the common denominator is usually the least common multiple (LCM) of $$x$$ and $$y$$. 3. **Find the common denominator of $$\frac{1}{a-1}$$ and $$\frac{1}{a+1}$$:** Since $$a-1$$ and $$a+1$$ are distinct linear factors, their LCM is simply their product: $$\text{LCM} = (a-1)(a+1)$$ 4. **Rewrite each fraction with the common denominator:** $$\frac{1}{a-1} = \frac{a+1}{(a-1)(a+1)}$$ $$\frac{1}{a+1} = \frac{a-1}{(a-1)(a+1)}$$ 5. **Subtract the two fractions:** $$\frac{a+1}{(a-1)(a+1)} - \frac{a-1}{(a-1)(a+1)} = \frac{a+1 - (a-1)}{(a-1)(a+1)}$$ 6. **Simplify the numerator:** $$a+1 - (a-1) = a + 1 - a + 1 = 2$$ 7. **Final simplified expression:** $$\frac{2}{(a-1)(a+1)}$$ **Answer:** The common denominator of $$\frac{1}{a-1}$$ and $$\frac{1}{a+1}$$ is $$(a-1)(a+1)$$, and their difference simplifies to $$\frac{2}{(a-1)(a+1)}$$.