x+6=(6+3x)(6+3x)

Solution for x+6=(6+3x)(6+3x) equation:

Simplifying
x + 6 = (6 + 3x)(6 + 3x)

Reorder the terms:
6 + x = (6 + 3x)(6 + 3x)

Multiply (6 + 3x) * (6 + 3x)
6 + x = (6(6 + 3x) + 3x * (6 + 3x))
6 + x = ((6 * 6 + 3x * 6) + 3x * (6 + 3x))
6 + x = ((36 + 18x) + 3x * (6 + 3x))
6 + x = (36 + 18x + (6 * 3x + 3x * 3x))
6 + x = (36 + 18x + (18x + 9x2))

Combine like terms: 18x + 18x = 36x
6 + x = (36 + 36x + 9x2)

Solving
6 + x = 36 + 36x + 9x2

Solving for variable 'x'.

Reorder the terms:
6 + -36 + x + -36x + -9x2 = 36 + 36x + 9x2 + -36 + -36x + -9x2

Combine like terms: 6 + -36 = -30
-30 + x + -36x + -9x2 = 36 + 36x + 9x2 + -36 + -36x + -9x2

Combine like terms: x + -36x = -35x
-30 + -35x + -9x2 = 36 + 36x + 9x2 + -36 + -36x + -9x2

Reorder the terms:
-30 + -35x + -9x2 = 36 + -36 + 36x + -36x + 9x2 + -9x2

Combine like terms: 36 + -36 = 0
-30 + -35x + -9x2 = 0 + 36x + -36x + 9x2 + -9x2
-30 + -35x + -9x2 = 36x + -36x + 9x2 + -9x2

Combine like terms: 36x + -36x = 0
-30 + -35x + -9x2 = 0 + 9x2 + -9x2
-30 + -35x + -9x2 = 9x2 + -9x2

Combine like terms: 9x2 + -9x2 = 0
-30 + -35x + -9x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(30 + 35x + 9x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(30 + 35x + 9x2)' equal to zero and attempt to solve:

Simplifying
30 + 35x + 9x2 = 0

Solving
30 + 35x + 9x2 = 0

Begin completing the square.  Divide all terms by
9 the coefficient of the squared term: 

Divide each side by '9'.
3.333333333 + 3.888888889x + x2 = 0

Move the constant term to the right:

Add '-3.333333333' to each side of the equation.
3.333333333 + 3.888888889x + -3.333333333 + x2 = 0 + -3.333333333

Reorder the terms:
3.333333333 + -3.333333333 + 3.888888889x + x2 = 0 + -3.333333333

Combine like terms: 3.333333333 + -3.333333333 = 0.000000000
0.000000000 + 3.888888889x + x2 = 0 + -3.333333333
3.888888889x + x2 = 0 + -3.333333333

Combine like terms: 0 + -3.333333333 = -3.333333333
3.888888889x + x2 = -3.333333333

The x term is 3.888888889x.  Take half its coefficient (1.944444445).
Square it (3.780864200) and add it to both sides.

Add '3.780864200' to each side of the equation.
3.888888889x + 3.780864200 + x2 = -3.333333333 + 3.780864200

Reorder the terms:
3.780864200 + 3.888888889x + x2 = -3.333333333 + 3.780864200

Combine like terms: -3.333333333 + 3.780864200 = 0.447530867
3.780864200 + 3.888888889x + x2 = 0.447530867

Factor a perfect square on the left side:
(x + 1.944444445)(x + 1.944444445) = 0.447530867

Calculate the square root of the right side: 0.668977479

Break this problem into two subproblems by setting 
(x + 1.944444445) equal to 0.668977479 and -0.668977479.

Subproblem 1
x + 1.944444445 = 0.668977479

Simplifying
x + 1.944444445 = 0.668977479

Reorder the terms:
1.944444445 + x = 0.668977479

Solving
1.944444445 + x = 0.668977479

Solving for variable 'x'.

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

Add '-1.944444445' to each side of the equation.
1.944444445 + -1.944444445 + x = 0.668977479 + -1.944444445

Combine like terms: 1.944444445 + -1.944444445 = 0.000000000
0.000000000 + x = 0.668977479 + -1.944444445
x = 0.668977479 + -1.944444445

Combine like terms: 0.668977479 + -1.944444445 = -1.275466966
x = -1.275466966

Simplifying
x = -1.275466966

Subproblem 2
x + 1.944444445 = -0.668977479

Simplifying
x + 1.944444445 = -0.668977479

Reorder the terms:
1.944444445 + x = -0.668977479

Solving
1.944444445 + x = -0.668977479

Solving for variable 'x'.

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

Add '-1.944444445' to each side of the equation.
1.944444445 + -1.944444445 + x = -0.668977479 + -1.944444445

Combine like terms: 1.944444445 + -1.944444445 = 0.000000000
0.000000000 + x = -0.668977479 + -1.944444445
x = -0.668977479 + -1.944444445

Combine like terms: -0.668977479 + -1.944444445 = -2.613421924
x = -2.613421924

Simplifying
x = -2.613421924

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.275466966, -2.613421924}
Solution
x = {-1.275466966, -2.613421924}