Here you have 2 constraints

x + y + 2z = 2 call it f

z-x^2 -y^2 = 0 call it g

You want to maxmimize the distance to the origin

(maximize the square of the distance)

Call it D = x^2 + y^2 + z^2

gradD= L1 gradf + L2 gradg

This leads to the equations:

1) 2x = L1 -2xL2

2) 2y = L1 -2yL2

3) 2z= 2L1 + L2

4) x + y + 2z =2

5) z = x^2 + y^2

Now

i Eliminate L2 between eqns 1) and 2) and you obtain x = y

ii Use this in 4) to get z = 1 - x

iii Use this in 5 to get x = 1/2 or -1

Use this to finish