To solve these equations, we need to isolate the square root term on one side of the equation and then square both sides to eliminate the square root.

1) √x + 1 = 1
Subtract 1 from both sides:
√x = 0
Square both sides:
(√x)^2 = 0^2
x = 0

2) √x + 1 = 2
Subtract 1 from both sides:
√x = 1
Square both sides:
(√x)^2 = 1^2
x = 1

3) √x + 1 = -2
Subtract 1 from both sides:
√x = -3
This equation has no real solutions since the square root of a number cannot be negative.

4) √x + 1 = -1
Subtract 1 from both sides:
√x = -2
This equation has no real solutions since the square root of a number cannot be negative.

So, the solutions are x = 0 and x = 1.