Find the Cartesian equation of the locus of z given by |z+1|+|z-1|=4?

Where are you stuck? Algebraic solution, drawing by hand, or some other point?

Let's assume it's the algebraic solution. You have three cases here:

(1) z+1, z-1 both <= 0. This would mean that z <= -1. In this case,

-(z+1) + -(z-1) = 4

Solve; is this z value consistent with the branch assumption that z <= -1 ? If so, then this solution is part of the locus.

(2) z+1, z-1 both >= 0. This means z >= 1, and (z+1)+(z-1) = 4. Treat similarly.

(3) z-1 < 0, z+1 > 0. This means that -1 < z < 1. Treat similarly.

Now, take the *union* of each of your solutions. That is your solution locus.