What is the solution set for x in the equation below square root of x + 1 -1 = x?

The notation in your question is not clear, but I presume that the equation is:
√(x + 1) - 1 = x

In order to solve irrational equation, you need to separate the radicand from the rest:
√(x + 1) = x + 1

Now, you need to set the condition in order to have a positive number as a radicand (you cannot have a square root of a negative number):
(x + 1) ≥ 0
x ≥ -1

Once the condition is set, you can elevate everything to the second power:
(√(x + 1))² = (x + 1)²
x + 1 = x² + 2x + 1
x² + x = 0
x(x + 1) = 0
x = 0  and x = -1 

Now that you have the solutions, check if they are relevant according to the condition you have previously found: in this case, yes, they are both greater than or equal to -1.

Therefore, the set of solutions is {-1, 0}