Let's simplify the expression step by step:

Given expression: (\sqrt{x + 2\sqrt{x - 1}} - \sqrt{x - 2\sqrt{x - 1}})

Step 1: Let (a = \sqrt{x - 1}).

This transforms the given expression into:
(\sqrt{a^2 + 2a} - \sqrt{a^2 - 2a})

Step 2: Simplify further by substituting the values:
(\sqrt{(a + 1)^2} - \sqrt{(a - 1)^2})

Step 3: Simplify the square roots:
(a + 1 - (a - 1))

Step 4: Combine like terms:
(a + 1 - a + 1 = 2)

Therefore, the simplified expression is (2).