4(x-1)=2(x+3)(2x-3)

Solution for 4(x-1)=2(x+3)(2x-3) equation:

Simplifying
4(x + -1) = 2(x + 3)(2x + -3)

Reorder the terms:
4(-1 + x) = 2(x + 3)(2x + -3)
(-1 * 4 + x * 4) = 2(x + 3)(2x + -3)
(-4 + 4x) = 2(x + 3)(2x + -3)

Reorder the terms:
-4 + 4x = 2(3 + x)(2x + -3)

Reorder the terms:
-4 + 4x = 2(3 + x)(-3 + 2x)

Multiply (3 + x) * (-3 + 2x)
-4 + 4x = 2(3(-3 + 2x) + x(-3 + 2x))
-4 + 4x = 2((-3 * 3 + 2x * 3) + x(-3 + 2x))
-4 + 4x = 2((-9 + 6x) + x(-3 + 2x))
-4 + 4x = 2(-9 + 6x + (-3 * x + 2x * x))
-4 + 4x = 2(-9 + 6x + (-3x + 2x2))

Combine like terms: 6x + -3x = 3x
-4 + 4x = 2(-9 + 3x + 2x2)
-4 + 4x = (-9 * 2 + 3x * 2 + 2x2 * 2)
-4 + 4x = (-18 + 6x + 4x2)

Solving
-4 + 4x = -18 + 6x + 4x2

Solving for variable 'x'.

Reorder the terms:
-4 + 18 + 4x + -6x + -4x2 = -18 + 6x + 4x2 + 18 + -6x + -4x2

Combine like terms: -4 + 18 = 14
14 + 4x + -6x + -4x2 = -18 + 6x + 4x2 + 18 + -6x + -4x2

Combine like terms: 4x + -6x = -2x
14 + -2x + -4x2 = -18 + 6x + 4x2 + 18 + -6x + -4x2

Reorder the terms:
14 + -2x + -4x2 = -18 + 18 + 6x + -6x + 4x2 + -4x2

Combine like terms: -18 + 18 = 0
14 + -2x + -4x2 = 0 + 6x + -6x + 4x2 + -4x2
14 + -2x + -4x2 = 6x + -6x + 4x2 + -4x2

Combine like terms: 6x + -6x = 0
14 + -2x + -4x2 = 0 + 4x2 + -4x2
14 + -2x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
14 + -2x + -4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(7 + -1x + -2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(7 + -1x + -2x2)' equal to zero and attempt to solve:

Simplifying
7 + -1x + -2x2 = 0

Solving
7 + -1x + -2x2 = 0

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

Divide each side by '-2'.
-3.5 + 0.5x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 0.5x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 0.5x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 0.5x + x2 = 0 + 3.5
0.5x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
0.5x + x2 = 3.5

The x term is 0.5x.  Take half its coefficient (0.25).
Square it (0.0625) and add it to both sides.

Add '0.0625' to each side of the equation.
0.5x + 0.0625 + x2 = 3.5 + 0.0625

Reorder the terms:
0.0625 + 0.5x + x2 = 3.5 + 0.0625

Combine like terms: 3.5 + 0.0625 = 3.5625
0.0625 + 0.5x + x2 = 3.5625

Factor a perfect square on the left side:
(x + 0.25)(x + 0.25) = 3.5625

Calculate the square root of the right side: 1.887458609

Break this problem into two subproblems by setting 
(x + 0.25) equal to 1.887458609 and -1.887458609.

Subproblem 1
x + 0.25 = 1.887458609

Simplifying
x + 0.25 = 1.887458609

Reorder the terms:
0.25 + x = 1.887458609

Solving
0.25 + x = 1.887458609

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = 1.887458609 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = 1.887458609 + -0.25
x = 1.887458609 + -0.25

Combine like terms: 1.887458609 + -0.25 = 1.637458609
x = 1.637458609

Simplifying
x = 1.637458609

Subproblem 2
x + 0.25 = -1.887458609

Simplifying
x + 0.25 = -1.887458609

Reorder the terms:
0.25 + x = -1.887458609

Solving
0.25 + x = -1.887458609

Solving for variable 'x'.

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

Add '-0.25' to each side of the equation.
0.25 + -0.25 + x = -1.887458609 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + x = -1.887458609 + -0.25
x = -1.887458609 + -0.25

Combine like terms: -1.887458609 + -0.25 = -2.137458609
x = -2.137458609

Simplifying
x = -2.137458609

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.637458609, -2.137458609}
Solution
x = {1.637458609, -2.137458609}