Choice A is the correct answer.

1. Take the square root of both sides: $x - 1 = \pm \sqrt{5}$.
2. Add 1 to both sides to isolate x: $x = 1 \pm \sqrt{5}$.

This represents two irrational solutions: $1 + \sqrt{5}$ and $1 - \sqrt{5}$.

Choice B is incorrect because of a sign error (subtracted 1 instead of adding). Choice C is incorrect because it adds 1 inside the radical, which is invalid. Choice D is incorrect because it swaps the order of operations.