3x(x+1)+x(3x-6)=8x+1

Solution for 3x(x+1)+x(3x-6)=8x+1 equation:

Simplifying
3x(x + 1) + x(3x + -6) = 8x + 1

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

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

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

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

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

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

Solving
-3x + 6x2 = 1 + 8x

Solving for variable 'x'.

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

Combine like terms: -3x + -8x = -11x
-1 + -11x + 6x2 = 1 + 8x + -1 + -8x

Reorder the terms:
-1 + -11x + 6x2 = 1 + -1 + 8x + -8x

Combine like terms: 1 + -1 = 0
-1 + -11x + 6x2 = 0 + 8x + -8x
-1 + -11x + 6x2 = 8x + -8x

Combine like terms: 8x + -8x = 0
-1 + -11x + 6x2 = 0

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

Divide each side by '6'.
-0.1666666667 + -1.833333333x + x2 = 0

Move the constant term to the right:

Add '0.1666666667' to each side of the equation.
-0.1666666667 + -1.833333333x + 0.1666666667 + x2 = 0 + 0.1666666667

Reorder the terms:
-0.1666666667 + 0.1666666667 + -1.833333333x + x2 = 0 + 0.1666666667

Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000
0.0000000000 + -1.833333333x + x2 = 0 + 0.1666666667
-1.833333333x + x2 = 0 + 0.1666666667

Combine like terms: 0 + 0.1666666667 = 0.1666666667
-1.833333333x + x2 = 0.1666666667

The x term is -1.833333333x.  Take half its coefficient (-0.9166666665).
Square it (0.8402777775) and add it to both sides.

Add '0.8402777775' to each side of the equation.
-1.833333333x + 0.8402777775 + x2 = 0.1666666667 + 0.8402777775

Reorder the terms:
0.8402777775 + -1.833333333x + x2 = 0.1666666667 + 0.8402777775

Combine like terms: 0.1666666667 + 0.8402777775 = 1.0069444442
0.8402777775 + -1.833333333x + x2 = 1.0069444442

Factor a perfect square on the left side:
(x + -0.9166666665)(x + -0.9166666665) = 1.0069444442

Calculate the square root of the right side: 1.003466215

Break this problem into two subproblems by setting 
(x + -0.9166666665) equal to 1.003466215 and -1.003466215.

Subproblem 1
x + -0.9166666665 = 1.003466215

Simplifying
x + -0.9166666665 = 1.003466215

Reorder the terms:
-0.9166666665 + x = 1.003466215

Solving
-0.9166666665 + x = 1.003466215

Solving for variable 'x'.

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

Add '0.9166666665' to each side of the equation.
-0.9166666665 + 0.9166666665 + x = 1.003466215 + 0.9166666665

Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000
0.0000000000 + x = 1.003466215 + 0.9166666665
x = 1.003466215 + 0.9166666665

Combine like terms: 1.003466215 + 0.9166666665 = 1.9201328815
x = 1.9201328815

Simplifying
x = 1.9201328815

Subproblem 2
x + -0.9166666665 = -1.003466215

Simplifying
x + -0.9166666665 = -1.003466215

Reorder the terms:
-0.9166666665 + x = -1.003466215

Solving
-0.9166666665 + x = -1.003466215

Solving for variable 'x'.

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

Add '0.9166666665' to each side of the equation.
-0.9166666665 + 0.9166666665 + x = -1.003466215 + 0.9166666665

Combine like terms: -0.9166666665 + 0.9166666665 = 0.0000000000
0.0000000000 + x = -1.003466215 + 0.9166666665
x = -1.003466215 + 0.9166666665

Combine like terms: -1.003466215 + 0.9166666665 = -0.0867995485
x = -0.0867995485

Simplifying
x = -0.0867995485

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.9201328815, -0.0867995485}