If you'd be interested to know where the "false solution" came from, it was introduced in the step where ThePerfectHacker took the square of both sides.

Example: x = 3. Of course, this 'equation' is already 'solved', but suppose I square both sides: x² = 9. Now, I have introduced the 'solution' x = -3, which isn't a solution to the initial equation.

If you're a bit uncomfortable with just squaring, possibly introducing false solutions, and in the end checking your answers, you can do the following: squaring is allowed if both sides have the same sign. Now in that step, we had:

$\displaystyle 2\sqrt{2x+1}=x+1$

But you know that (the double of) a square root is nonnegative, so the right hand side can't be negative either. This yields the condition: $\displaystyle x\ge -1$. If you take along this condition, all your steps remain equivalent equations and you'll see that the second solution is smaller than -1, hence you have to drop it.

Somtimes this is more work than just checking your answers, sometimes it isn't. In any case, it's 'elegant' in the way that you keep working with equivalent equation, not introducing any solutions because of the acquired condition on x.

I hope this at least 'clarifies' what's happenning here, mathemtically