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

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

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

Reorder the terms:
4x(1 + 2x) = 3(x + 2)
(1 * 4x + 2x * 4x) = 3(x + 2)
(4x + 8x2) = 3(x + 2)

Reorder the terms:
4x + 8x2 = 3(2 + x)
4x + 8x2 = (2 * 3 + x * 3)
4x + 8x2 = (6 + 3x)

Solving
4x + 8x2 = 6 + 3x

Solving for variable 'x'.

Reorder the terms:
-6 + 4x + -3x + 8x2 = 6 + 3x + -6 + -3x

Combine like terms: 4x + -3x = 1x
-6 + 1x + 8x2 = 6 + 3x + -6 + -3x

Reorder the terms:
-6 + 1x + 8x2 = 6 + -6 + 3x + -3x

Combine like terms: 6 + -6 = 0
-6 + 1x + 8x2 = 0 + 3x + -3x
-6 + 1x + 8x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-6 + 1x + 8x2 = 0

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

Divide each side by '8'.
-0.75 + 0.125x + x2 = 0

Move the constant term to the right:

Add '0.75' to each side of the equation.
-0.75 + 0.125x + 0.75 + x2 = 0 + 0.75

Reorder the terms:
-0.75 + 0.75 + 0.125x + x2 = 0 + 0.75

Combine like terms: -0.75 + 0.75 = 0.00
0.00 + 0.125x + x2 = 0 + 0.75
0.125x + x2 = 0 + 0.75

Combine like terms: 0 + 0.75 = 0.75
0.125x + x2 = 0.75

The x term is 0.125x.  Take half its coefficient (0.0625).
Square it (0.00390625) and add it to both sides.

Add '0.00390625' to each side of the equation.
0.125x + 0.00390625 + x2 = 0.75 + 0.00390625

Reorder the terms:
0.00390625 + 0.125x + x2 = 0.75 + 0.00390625

Combine like terms: 0.75 + 0.00390625 = 0.75390625
0.00390625 + 0.125x + x2 = 0.75390625

Factor a perfect square on the left side:
(x + 0.0625)(x + 0.0625) = 0.75390625

Calculate the square root of the right side: 0.868277749

Break this problem into two subproblems by setting 
(x + 0.0625) equal to 0.868277749 and -0.868277749.

Subproblem 1
x + 0.0625 = 0.868277749

Simplifying
x + 0.0625 = 0.868277749

Reorder the terms:
0.0625 + x = 0.868277749

Solving
0.0625 + x = 0.868277749

Solving for variable 'x'.

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

Add '-0.0625' to each side of the equation.
0.0625 + -0.0625 + x = 0.868277749 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + x = 0.868277749 + -0.0625
x = 0.868277749 + -0.0625

Combine like terms: 0.868277749 + -0.0625 = 0.805777749
x = 0.805777749

Simplifying
x = 0.805777749

Subproblem 2
x + 0.0625 = -0.868277749

Simplifying
x + 0.0625 = -0.868277749

Reorder the terms:
0.0625 + x = -0.868277749

Solving
0.0625 + x = -0.868277749

Solving for variable 'x'.

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

Add '-0.0625' to each side of the equation.
0.0625 + -0.0625 + x = -0.868277749 + -0.0625

Combine like terms: 0.0625 + -0.0625 = 0.0000
0.0000 + x = -0.868277749 + -0.0625
x = -0.868277749 + -0.0625

Combine like terms: -0.868277749 + -0.0625 = -0.930777749
x = -0.930777749

Simplifying
x = -0.930777749

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.805777749, -0.930777749}