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

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

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

Reorder the terms:
(3 + x)(3x + -1) = 4(x + 2) + 1

Reorder the terms:
(3 + x)(-1 + 3x) = 4(x + 2) + 1

Multiply (3 + x) * (-1 + 3x)
(3(-1 + 3x) + x(-1 + 3x)) = 4(x + 2) + 1
((-1 * 3 + 3x * 3) + x(-1 + 3x)) = 4(x + 2) + 1
((-3 + 9x) + x(-1 + 3x)) = 4(x + 2) + 1
(-3 + 9x + (-1 * x + 3x * x)) = 4(x + 2) + 1
(-3 + 9x + (-1x + 3x2)) = 4(x + 2) + 1

Combine like terms: 9x + -1x = 8x
(-3 + 8x + 3x2) = 4(x + 2) + 1

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

Reorder the terms:
-3 + 8x + 3x2 = 8 + 1 + 4x

Combine like terms: 8 + 1 = 9
-3 + 8x + 3x2 = 9 + 4x

Solving
-3 + 8x + 3x2 = 9 + 4x

Solving for variable 'x'.

Reorder the terms:
-3 + -9 + 8x + -4x + 3x2 = 9 + 4x + -9 + -4x

Combine like terms: -3 + -9 = -12
-12 + 8x + -4x + 3x2 = 9 + 4x + -9 + -4x

Combine like terms: 8x + -4x = 4x
-12 + 4x + 3x2 = 9 + 4x + -9 + -4x

Reorder the terms:
-12 + 4x + 3x2 = 9 + -9 + 4x + -4x

Combine like terms: 9 + -9 = 0
-12 + 4x + 3x2 = 0 + 4x + -4x
-12 + 4x + 3x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-12 + 4x + 3x2 = 0

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

Divide each side by '3'.
-4 + 1.333333333x + x2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 1.333333333x + 4 + x2 = 0 + 4

Reorder the terms:
-4 + 4 + 1.333333333x + x2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 1.333333333x + x2 = 0 + 4
1.333333333x + x2 = 0 + 4

Combine like terms: 0 + 4 = 4
1.333333333x + x2 = 4

The x term is 1.333333333x.  Take half its coefficient (0.6666666665).
Square it (0.4444444442) and add it to both sides.

Add '0.4444444442' to each side of the equation.
1.333333333x + 0.4444444442 + x2 = 4 + 0.4444444442

Reorder the terms:
0.4444444442 + 1.333333333x + x2 = 4 + 0.4444444442

Combine like terms: 4 + 0.4444444442 = 4.4444444442
0.4444444442 + 1.333333333x + x2 = 4.4444444442

Factor a perfect square on the left side:
(x + 0.6666666665)(x + 0.6666666665) = 4.4444444442

Calculate the square root of the right side: 2.108185107

Break this problem into two subproblems by setting 
(x + 0.6666666665) equal to 2.108185107 and -2.108185107.

Subproblem 1
x + 0.6666666665 = 2.108185107

Simplifying
x + 0.6666666665 = 2.108185107

Reorder the terms:
0.6666666665 + x = 2.108185107

Solving
0.6666666665 + x = 2.108185107

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = 2.108185107 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = 2.108185107 + -0.6666666665
x = 2.108185107 + -0.6666666665

Combine like terms: 2.108185107 + -0.6666666665 = 1.4415184405
x = 1.4415184405

Simplifying
x = 1.4415184405

Subproblem 2
x + 0.6666666665 = -2.108185107

Simplifying
x + 0.6666666665 = -2.108185107

Reorder the terms:
0.6666666665 + x = -2.108185107

Solving
0.6666666665 + x = -2.108185107

Solving for variable 'x'.

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

Add '-0.6666666665' to each side of the equation.
0.6666666665 + -0.6666666665 + x = -2.108185107 + -0.6666666665

Combine like terms: 0.6666666665 + -0.6666666665 = 0.0000000000
0.0000000000 + x = -2.108185107 + -0.6666666665
x = -2.108185107 + -0.6666666665

Combine like terms: -2.108185107 + -0.6666666665 = -2.7748517735
x = -2.7748517735

Simplifying
x = -2.7748517735

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4415184405, -2.7748517735}