ЯсноПонятно24

Given the equation x^2 + 2xz + 4xy + 4y^2 + 4yz + z^2 + sqrt(x+y+2z) + |x+1| = 0, we can simplify it by removing the absolute value.

Since |x+1| is always positive, it can be rewritten as x+1.

Therefore, the equation becomes: x^2 + 2xz + 4xy + 4y^2 + 4yz + z^2 + sqrt(x+y+2z) + x + 1 = 0

Now, let's group the terms: (x^2 + 4xy + x) + (2xz + 4yz) + (4y^2 + z^2) + sqrt(x+y+2z) + 1 = 0

Now, we can rewrite the equation as: (x + 1)^2 + 2z(x + 2y) + (2y + z)^2 + sqrt(x + y + 2z) + 1 = 0

Since the sum of squares is always non-negative, the only way for the equation to be equal to zero is if each term is equal to zero.

Therefore, we have: x + 1 = 0 x = -1

2z(x + 2y) = 0 2z(-1 + 2y) = 0 -2z + 4yz = 0 z(4y - 2) = 0 z = 0 or y = 1/2

(2y + z)^2 = 0 (2y + 0)^2 = 0 4y^2 = 0 y = 0

sqrt(x + y + 2z) = 0 sqrt(-1 + 0 + 2(0)) = 0 sqrt(-1) = 0 This is not possible.

Therefore, the only solution is x = -1, y = 0, z = 0.

Therefore, x + y + z = -1 + 0 + 0 = -1.