Let's work through finding the Least Common Denominator (LCD) for each of the rational expressions in the equations provided. The LCD is the least common multiple of the denominators involved.

1. Equation (a): $\frac{x-3}{x^2-25} + \frac{1}{x-5}$ The denominators are $x^2 - 25$ and $x - 5$. Factor $x^2 - 25$: $x^2 - 25 = (x-5)(x+5)$ So, the LCD is $(x-5)(x+5)$.

2. Equation (b): $\frac{x^2}{x-1} = \frac{1}{2}$ The only denominator here is $x - 1$, so the LCD is $x - 1$.

3. Equation (c): $\frac{4x + 1}{x - 1} - \frac{3}{x - 2} = \frac{12}{x^2 - 3x + 2}$ Factor the denominator $x^2 - 3x + 2$: $x^2 - 3x + 2 = (x - 1)(x - 2)$ The denominators are $x - 1$, $x - 2$, and $(x - 1)(x - 2)$. Therefore, the LCD is $(x - 1)(x - 2)$.

4. Equation (d): $\frac{1}{x - 4} + \frac{1}{x + 12} = \frac{1}{x - 2}$ The denominators are $x - 4$, $x + 12$, and $x - 2$. The LCD is $(x - 4)(x + 12)(x - 2)$.

5. Equation (e): $\frac{x - 6}{x^2 - 4x - 12} + \frac{2}{x + 12} = \frac{1}{x - 6}$ Factor $x^2 - 4x - 12$: $x^2 - 4x - 12 = (x - 6)(x + 2)$ The denominators are $(x - 6)(x + 2)$, $x + 12$, and $x - 6$. The LCD is $(x - 6)(x + 2)(x + 12)$.

Would you like more detailed steps for any specific part of this process?

Here are some related questions to consider:

1. How do you factor quadratic expressions like $x^2 - 25$ or $x^2 - 4x - 12$?
2. What is the importance of finding the LCD in solving rational equations?
3. How do you simplify expressions once you have the LCD?
4. What are some common mistakes to avoid when finding the LCD?
5. How can you verify that your calculated LCD is correct?

Tip: Always double-check the factorization of the denominators to ensure accuracy when finding the LCD.