1) √х - 1 = х + 3
Square both sides to eliminate the square root:
(√x - 1)² = (x + 3)²
x - 2√x + 1 = x² + 6x + 9

Rearrange the equation to put it in standard form:
x² + 5x + 8 = 0
This is a quadratic equation that can be solved using the quadratic formula. The solutions are:
x = (-5 ± √(5² - 418)) / 2*1
x = (-5 ± √(25 - 32)) / 2
x = (-5 ± √(-7)) / 2
x = (-5 ± i√7) / 2

Therefore, the solutions to the equation √x - 1 = x + 3 are x = (-5 + i√7) / 2 and x = (-5 - i√7) / 2.

2) √x + 4 = √x - 6
Subtract √x from both sides:
4 = -6

There is no solution to this equation since 4 cannot equal -6.

3) ³√x + 2 = x - 1
Cube both sides to eliminate the cube root:
(³√x + 2)³ = (x - 1)³
x + 6√(x)² + 12√(x) + 8 = x³ - 3x² + 3x - 1

Rearrange the equation to put it in standard form:
x³ - 2x² + 3x - 9 = 0
This is a cubic equation that can be solved using various methods, such as factoring or using Cardano's formula. The solutions are finite and can be found by solving the cubic equation.

Therefore, the solution(s) to the equation ³√x + 2 = x - 1 are complex and can be determined by solving the cubic equation x³ - 2x² + 3x - 9 = 0.