4x(2x+1)-2x=34

Solution for 4x(2x+1)-2x=34 equation:

Simplifying
4x(2x + 1) + -2x = 34

Reorder the terms:
4x(1 + 2x) + -2x = 34
(1 * 4x + 2x * 4x) + -2x = 34
(4x + 8x2) + -2x = 34

Reorder the terms:
4x + -2x + 8x2 = 34

Combine like terms: 4x + -2x = 2x
2x + 8x2 = 34

Solving
2x + 8x2 = 34

Solving for variable 'x'.

Reorder the terms:
-34 + 2x + 8x2 = 34 + -34

Combine like terms: 34 + -34 = 0
-34 + 2x + 8x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-17 + x + 4x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-17 + x + 4x2)' equal to zero and attempt to solve:

Simplifying
-17 + x + 4x2 = 0

Solving
-17 + x + 4x2 = 0

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

Divide each side by '4'.
-4.25 + 0.25x + x2 = 0

Move the constant term to the right:

Add '4.25' to each side of the equation.
-4.25 + 0.25x + 4.25 + x2 = 0 + 4.25

Reorder the terms:
-4.25 + 4.25 + 0.25x + x2 = 0 + 4.25

Combine like terms: -4.25 + 4.25 = 0.00
0.00 + 0.25x + x2 = 0 + 4.25
0.25x + x2 = 0 + 4.25

Combine like terms: 0 + 4.25 = 4.25
0.25x + x2 = 4.25

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.
0.25x + 0.25 + x2 = 4.25 + 0.25

Reorder the terms:
0.25 + 0.25x + x2 = 4.25 + 0.25

Combine like terms: 4.25 + 0.25 = 4.5
0.25 + 0.25x + x2 = 4.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 4.5

Calculate the square root of the right side: 2.121320344

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.121320344 and -2.121320344.

Subproblem 1
x + 0.5 = 2.121320344

Simplifying
x + 0.5 = 2.121320344

Reorder the terms:
0.5 + x = 2.121320344

Solving
0.5 + x = 2.121320344

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 = 2.121320344 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.121320344 + -0.5
x = 2.121320344 + -0.5

Combine like terms: 2.121320344 + -0.5 = 1.621320344
x = 1.621320344

Simplifying
x = 1.621320344

Subproblem 2
x + 0.5 = -2.121320344

Simplifying
x + 0.5 = -2.121320344

Reorder the terms:
0.5 + x = -2.121320344

Solving
0.5 + x = -2.121320344

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 = -2.121320344 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.121320344 + -0.5
x = -2.121320344 + -0.5

Combine like terms: -2.121320344 + -0.5 = -2.621320344
x = -2.621320344

Simplifying
x = -2.621320344

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.621320344, -2.621320344}
Solution
x = {1.621320344, -2.621320344}