-2x(x+6)=3(2x+3)

Solution for -2x(x+6)=3(2x+3) equation:

Simplifying
-2x(x + 6) = 3(2x + 3)

Reorder the terms:
-2x(6 + x) = 3(2x + 3)
(6 * -2x + x * -2x) = 3(2x + 3)
(-12x + -2x2) = 3(2x + 3)

Reorder the terms:
-12x + -2x2 = 3(3 + 2x)
-12x + -2x2 = (3 * 3 + 2x * 3)
-12x + -2x2 = (9 + 6x)

Solving
-12x + -2x2 = 9 + 6x

Solving for variable 'x'.

Reorder the terms:
-9 + -12x + -6x + -2x2 = 9 + 6x + -9 + -6x

Combine like terms: -12x + -6x = -18x
-9 + -18x + -2x2 = 9 + 6x + -9 + -6x

Reorder the terms:
-9 + -18x + -2x2 = 9 + -9 + 6x + -6x

Combine like terms: 9 + -9 = 0
-9 + -18x + -2x2 = 0 + 6x + -6x
-9 + -18x + -2x2 = 6x + -6x

Combine like terms: 6x + -6x = 0
-9 + -18x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(9 + 18x + 2x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(9 + 18x + 2x2)' equal to zero and attempt to solve:

Simplifying
9 + 18x + 2x2 = 0

Solving
9 + 18x + 2x2 = 0

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

Divide each side by '2'.
4.5 + 9x + x2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 9x + -4.5 + x2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 9x + x2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 9x + x2 = 0 + -4.5
9x + x2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
9x + x2 = -4.5

The x term is 9x.  Take half its coefficient (4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
9x + 20.25 + x2 = -4.5 + 20.25

Reorder the terms:
20.25 + 9x + x2 = -4.5 + 20.25

Combine like terms: -4.5 + 20.25 = 15.75
20.25 + 9x + x2 = 15.75

Factor a perfect square on the left side:
(x + 4.5)(x + 4.5) = 15.75

Calculate the square root of the right side: 3.968626967

Break this problem into two subproblems by setting 
(x + 4.5) equal to 3.968626967 and -3.968626967.

Subproblem 1
x + 4.5 = 3.968626967

Simplifying
x + 4.5 = 3.968626967

Reorder the terms:
4.5 + x = 3.968626967

Solving
4.5 + x = 3.968626967

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = 3.968626967 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = 3.968626967 + -4.5
x = 3.968626967 + -4.5

Combine like terms: 3.968626967 + -4.5 = -0.531373033
x = -0.531373033

Simplifying
x = -0.531373033

Subproblem 2
x + 4.5 = -3.968626967

Simplifying
x + 4.5 = -3.968626967

Reorder the terms:
4.5 + x = -3.968626967

Solving
4.5 + x = -3.968626967

Solving for variable 'x'.

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

Add '-4.5' to each side of the equation.
4.5 + -4.5 + x = -3.968626967 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + x = -3.968626967 + -4.5
x = -3.968626967 + -4.5

Combine like terms: -3.968626967 + -4.5 = -8.468626967
x = -8.468626967

Simplifying
x = -8.468626967

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.531373033, -8.468626967}
Solution
x = {-0.531373033, -8.468626967}