Solving $\sqrt{x+5} = x - 1$

At no stage...

Here is what happens, you squared the equation

$$\sqrt{x+5} =x-1 \,.$$

But then, if the two sides of the equation have the same absolute value, but opposite signs they are not equal in this equation but they become equal at the next step.

This means that the equation $x+5=(x-1)^2$ possibly has more solutions. So the answers you got at the end are no necessarily the solution, they are just the possible solutions.

You have to check which one works.

+1 for showing your work :)

The problem is: $$x+5 = (x - 1)^2$$ doesn't imply $$\sqrt{x + 5} = x - 1$$ but $$\sqrt{x + 5} = \vert x - 1\vert$$