Thanks for asking the question!

ANSWER::

I am going to SHOW you a very lazy method as you have given me options too. So , let's start

Question = √[2x² -1 + 2x√(x²-1)]

Let us check the answer with every OPTION::

1. Suppose option 1 is the answer

√[2x² -1 + 2x√(x²-1)] = X - √(x²-1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² - 1 - 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² -1 - 2x√(x²-1)

LHS is not equal to RHS so , 1st option is not the answer.

2. Suppose option 2 is the answer

√[2x² -1 + 2x√(x²-1)] = x + √(x²-1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² - 1 + 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² -1 + 2x√(x²-1)

LHS is equal to RHS so , option 2 is correct.

3. Suppose option 3 is the answer

√[2x² -1 + 2x√(x²-1)] = x + √(x²+1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² + 1 + 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² +1 + 2x√(x²-1)

LHS is not equal to RHS so , option 3 is not the answer.

4. Suppose option 4 is the answer

√[2x² -1 + 2x√(x²-1)] = x - √(x²+1)

Squaring both sides

2x² -1 + 2x√(x²-1) = x² + x² + 1 - 2x√(x²-1)

2x² -1 + 2x√(x²-1) = 2x² + 1 - 2x√(x²-1)

LHS is not equal to RHS so , option 4 is not the answer.

Basically , as we were applying option 2nd option came to be correct so , we can leave the other options and not check them if its a single choice question.

But , if it is a Multiple Choice question we have to check all options.

Hope it helps!