x(3x+2)8=456

Solution for x(3x+2)8=456 equation:

Simplifying
x(3x + 2) * 8 = 456

Reorder the terms:
x(2 + 3x) * 8 = 456

Reorder the terms for easier multiplication:
8x(2 + 3x) = 456
(2 * 8x + 3x * 8x) = 456
(16x + 24x2) = 456

Solving
16x + 24x2 = 456

Solving for variable 'x'.

Reorder the terms:
-456 + 16x + 24x2 = 456 + -456

Combine like terms: 456 + -456 = 0
-456 + 16x + 24x2 = 0

Factor out the Greatest Common Factor (GCF), '8'.
8(-57 + 2x + 3x2) = 0

Ignore the factor 8.
Subproblem 1
Set the factor '(-57 + 2x + 3x2)' equal to zero and attempt to solve:

Simplifying
-57 + 2x + 3x2 = 0

Solving
-57 + 2x + 3x2 = 0

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

Divide each side by '3'.
-19 + 0.6666666667x + x2 = 0

Move the constant term to the right:

Add '19' to each side of the equation.
-19 + 0.6666666667x + 19 + x2 = 0 + 19

Reorder the terms:
-19 + 19 + 0.6666666667x + x2 = 0 + 19

Combine like terms: -19 + 19 = 0
0 + 0.6666666667x + x2 = 0 + 19
0.6666666667x + x2 = 0 + 19

Combine like terms: 0 + 19 = 19
0.6666666667x + x2 = 19

The x term is 0.6666666667x.  Take half its coefficient (0.3333333334).
Square it (0.1111111112) and add it to both sides.

Add '0.1111111112' to each side of the equation.
0.6666666667x + 0.1111111112 + x2 = 19 + 0.1111111112

Reorder the terms:
0.1111111112 + 0.6666666667x + x2 = 19 + 0.1111111112

Combine like terms: 19 + 0.1111111112 = 19.1111111112
0.1111111112 + 0.6666666667x + x2 = 19.1111111112

Factor a perfect square on the left side:
(x + 0.3333333334)(x + 0.3333333334) = 19.1111111112

Calculate the square root of the right side: 4.371625683

Break this problem into two subproblems by setting 
(x + 0.3333333334) equal to 4.371625683 and -4.371625683.

Subproblem 1
x + 0.3333333334 = 4.371625683

Simplifying
x + 0.3333333334 = 4.371625683

Reorder the terms:
0.3333333334 + x = 4.371625683

Solving
0.3333333334 + x = 4.371625683

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = 4.371625683 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = 4.371625683 + -0.3333333334
x = 4.371625683 + -0.3333333334

Combine like terms: 4.371625683 + -0.3333333334 = 4.0382923496
x = 4.0382923496

Simplifying
x = 4.0382923496

Subproblem 2
x + 0.3333333334 = -4.371625683

Simplifying
x + 0.3333333334 = -4.371625683

Reorder the terms:
0.3333333334 + x = -4.371625683

Solving
0.3333333334 + x = -4.371625683

Solving for variable 'x'.

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

Add '-0.3333333334' to each side of the equation.
0.3333333334 + -0.3333333334 + x = -4.371625683 + -0.3333333334

Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000
0.0000000000 + x = -4.371625683 + -0.3333333334
x = -4.371625683 + -0.3333333334

Combine like terms: -4.371625683 + -0.3333333334 = -4.7049590164
x = -4.7049590164

Simplifying
x = -4.7049590164

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.0382923496, -4.7049590164}
Solution
x = {4.0382923496, -4.7049590164}