(z+2)(z+2)=-2z

Solution for (z+2)(z+2)=-2z equation:

Simplifying
(z + 2)(z + 2) = -2z

Reorder the terms:
(2 + z)(z + 2) = -2z

Reorder the terms:
(2 + z)(2 + z) = -2z

Multiply (2 + z) * (2 + z)
(2(2 + z) + z(2 + z)) = -2z
((2 * 2 + z * 2) + z(2 + z)) = -2z
((4 + 2z) + z(2 + z)) = -2z
(4 + 2z + (2 * z + z * z)) = -2z
(4 + 2z + (2z + z2)) = -2z

Combine like terms: 2z + 2z = 4z
(4 + 4z + z2) = -2z

Solving
4 + 4z + z2 = -2z

Solving for variable 'z'.

Reorder the terms:
4 + 4z + 2z + z2 = -2z + 2z

Combine like terms: 4z + 2z = 6z
4 + 6z + z2 = -2z + 2z

Combine like terms: -2z + 2z = 0
4 + 6z + z2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-4' to each side of the equation.
4 + 6z + -4 + z2 = 0 + -4

Reorder the terms:
4 + -4 + 6z + z2 = 0 + -4

Combine like terms: 4 + -4 = 0
0 + 6z + z2 = 0 + -4
6z + z2 = 0 + -4

Combine like terms: 0 + -4 = -4
6z + z2 = -4

The z term is 6z.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6z + 9 + z2 = -4 + 9

Reorder the terms:
9 + 6z + z2 = -4 + 9

Combine like terms: -4 + 9 = 5
9 + 6z + z2 = 5

Factor a perfect square on the left side:
(z + 3)(z + 3) = 5

Calculate the square root of the right side: 2.236067978

Break this problem into two subproblems by setting 
(z + 3) equal to 2.236067978 and -2.236067978.

Subproblem 1
z + 3 = 2.236067978

Simplifying
z + 3 = 2.236067978

Reorder the terms:
3 + z = 2.236067978

Solving
3 + z = 2.236067978

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + z = 2.236067978 + -3

Combine like terms: 3 + -3 = 0
0 + z = 2.236067978 + -3
z = 2.236067978 + -3

Combine like terms: 2.236067978 + -3 = -0.763932022
z = -0.763932022

Simplifying
z = -0.763932022

Subproblem 2
z + 3 = -2.236067978

Simplifying
z + 3 = -2.236067978

Reorder the terms:
3 + z = -2.236067978

Solving
3 + z = -2.236067978

Solving for variable 'z'.

Move all terms containing z to the left, all other terms to the right.

Add '-3' to each side of the equation.
3 + -3 + z = -2.236067978 + -3

Combine like terms: 3 + -3 = 0
0 + z = -2.236067978 + -3
z = -2.236067978 + -3

Combine like terms: -2.236067978 + -3 = -5.236067978
z = -5.236067978

Simplifying
z = -5.236067978

Solution
The solution to the problem is based on the solutions
from the subproblems.
z = {-0.763932022, -5.236067978}