least common denominator
The least common denominator (LCD) is the smallest expression that every denominator in a group of fractions divides into evenly. It is the “smallest shared floor” on which all the fractions can stand together, so they can be added or compared.
For numbers it is just the least common multiple of the denominators — the LCD of 1/6 and 1/4 is 12. For rational expressions you build it from factors: factor every denominator, then take each distinct factor raised to the highest power it appears with anywhere. The product of those is the LCD.
Choosing the least common denominator rather than just any common denominator (such as the product of all the bottoms) keeps the resulting fraction as small as possible and saves you from heavy simplifying later. It is the efficient choice, not a different kind of answer — any common denominator gives a correct sum; the least one gives the cleanest.
Denominators (x - 1) and (x^2 - 1) = (x - 1)(x + 1): the LCD is (x - 1)(x + 1), not their product (x - 1)^2(x + 1).
Take each factor to its highest needed power — no more.