4x(x+1)-2x=32

Solution for 4x(x+1)-2x=32 equation:

Simplifying
4x(x + 1) + -2x = 32

Reorder the terms:
4x(1 + x) + -2x = 32
(1 * 4x + x * 4x) + -2x = 32
(4x + 4x2) + -2x = 32

Reorder the terms:
4x + -2x + 4x2 = 32

Combine like terms: 4x + -2x = 2x
2x + 4x2 = 32

Solving
2x + 4x2 = 32

Solving for variable 'x'.

Reorder the terms:
-32 + 2x + 4x2 = 32 + -32

Combine like terms: 32 + -32 = 0
-32 + 2x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-16 + x + 2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-16 + x + 2x2)' equal to zero and attempt to solve:

Simplifying
-16 + x + 2x2 = 0

Solving
-16 + x + 2x2 = 0

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

Divide each side by '2'.
-8 + 0.5x + x2 = 0

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 0.5x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + 0.5x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 0.5x + x2 = 0 + 8
0.5x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
0.5x + x2 = 8

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

Add '0.25' to each side of the equation.
0.5x + 0.25 + x2 = 8 + 0.25

Reorder the terms:
0.25 + 0.5x + x2 = 8 + 0.25

Combine like terms: 8 + 0.25 = 8.25
0.25 + 0.5x + x2 = 8.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.872281323 and -2.872281323.

Subproblem 1
x + 0.5 = 2.872281323

Simplifying
x + 0.5 = 2.872281323

Reorder the terms:
0.5 + x = 2.872281323

Solving
0.5 + x = 2.872281323

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 2.872281323 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.872281323 + -0.5
x = 2.872281323 + -0.5

Combine like terms: 2.872281323 + -0.5 = 2.372281323
x = 2.372281323

Simplifying
x = 2.372281323

Subproblem 2
x + 0.5 = -2.872281323

Simplifying
x + 0.5 = -2.872281323

Reorder the terms:
0.5 + x = -2.872281323

Solving
0.5 + x = -2.872281323

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -2.872281323 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.872281323 + -0.5
x = -2.872281323 + -0.5

Combine like terms: -2.872281323 + -0.5 = -3.372281323
x = -3.372281323

Simplifying
x = -3.372281323

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.372281323, -3.372281323}
Solution
x = {2.372281323, -3.372281323}