What mistake did I make and how do I fix it?

3/(x+1) = -1
3 = -x-1
x = -4

1/(x+1)+2/(x+1)= 3/(x+1)

You replaced 1/(x+1)+(2)/(x+1) with (x-1)+2(x+1) without changing the other side of the equation, and where did that (x-1) even come from?

You have two fractions with the same denominator of x+1. So you should multiply both sides by x+1:
1/(x+1) + 2/(x+1) = -1
1 + 2 = -1(x + 1)
3 = -x - 1

Add 1 to both sides:
4 = -x

Divide both sides by -1:
x = -4