This is kind of stupid since I already have the answers, but I feel like I overlooked an important aspect of Lagrange Multipliers to misuse it here.
Given $f = x^2 + xy + x - y$ and the constraint is the region formed by $y = 1 - x$ and the coordinate axis.
Now I realize I havce to test it alone the boundary, but could someone explain to me what I overlooked when I want to use Lagrange Multipliers?
I was thinking of apply Lagrange Multipliers to each constraint. So for instance on the line $y + x = g(x,y) = 1$ I would do
$\nabla f = \lambda \nabla g$
$g = 1$
After doing so I got a contradiction where x + y = 1 and x + y = 0