(x+5)(3x+4)=(2x+1)+(x+13)

Solution for (x+5)(3x+4)=(2x+1)+(x+13) equation:

Simplifying
(x + 5)(3x + 4) = (2x + 1) + (x + 13)

Reorder the terms:
(5 + x)(3x + 4) = (2x + 1) + (x + 13)

Reorder the terms:
(5 + x)(4 + 3x) = (2x + 1) + (x + 13)

Multiply (5 + x) * (4 + 3x)
(5(4 + 3x) + x(4 + 3x)) = (2x + 1) + (x + 13)
((4 * 5 + 3x * 5) + x(4 + 3x)) = (2x + 1) + (x + 13)
((20 + 15x) + x(4 + 3x)) = (2x + 1) + (x + 13)
(20 + 15x + (4 * x + 3x * x)) = (2x + 1) + (x + 13)
(20 + 15x + (4x + 3x2)) = (2x + 1) + (x + 13)

Combine like terms: 15x + 4x = 19x
(20 + 19x + 3x2) = (2x + 1) + (x + 13)

Reorder the terms:
20 + 19x + 3x2 = (1 + 2x) + (x + 13)

Remove parenthesis around (1 + 2x)
20 + 19x + 3x2 = 1 + 2x + (x + 13)

Reorder the terms:
20 + 19x + 3x2 = 1 + 2x + (13 + x)

Remove parenthesis around (13 + x)
20 + 19x + 3x2 = 1 + 2x + 13 + x

Reorder the terms:
20 + 19x + 3x2 = 1 + 13 + 2x + x

Combine like terms: 1 + 13 = 14
20 + 19x + 3x2 = 14 + 2x + x

Combine like terms: 2x + x = 3x
20 + 19x + 3x2 = 14 + 3x

Solving
20 + 19x + 3x2 = 14 + 3x

Solving for variable 'x'.

Reorder the terms:
20 + -14 + 19x + -3x + 3x2 = 14 + 3x + -14 + -3x

Combine like terms: 20 + -14 = 6
6 + 19x + -3x + 3x2 = 14 + 3x + -14 + -3x

Combine like terms: 19x + -3x = 16x
6 + 16x + 3x2 = 14 + 3x + -14 + -3x

Reorder the terms:
6 + 16x + 3x2 = 14 + -14 + 3x + -3x

Combine like terms: 14 + -14 = 0
6 + 16x + 3x2 = 0 + 3x + -3x
6 + 16x + 3x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
6 + 16x + 3x2 = 0

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

Divide each side by '3'.
2 + 5.333333333x + x2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 5.333333333x + -2 + x2 = 0 + -2

Reorder the terms:
2 + -2 + 5.333333333x + x2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 5.333333333x + x2 = 0 + -2
5.333333333x + x2 = 0 + -2

Combine like terms: 0 + -2 = -2
5.333333333x + x2 = -2

The x term is 5.333333333x.  Take half its coefficient (2.666666667).
Square it (7.111111113) and add it to both sides.

Add '7.111111113' to each side of the equation.
5.333333333x + 7.111111113 + x2 = -2 + 7.111111113

Reorder the terms:
7.111111113 + 5.333333333x + x2 = -2 + 7.111111113

Combine like terms: -2 + 7.111111113 = 5.111111113
7.111111113 + 5.333333333x + x2 = 5.111111113

Factor a perfect square on the left side:
(x + 2.666666667)(x + 2.666666667) = 5.111111113

Calculate the square root of the right side: 2.260776661

Break this problem into two subproblems by setting 
(x + 2.666666667) equal to 2.260776661 and -2.260776661.

Subproblem 1
x + 2.666666667 = 2.260776661

Simplifying
x + 2.666666667 = 2.260776661

Reorder the terms:
2.666666667 + x = 2.260776661

Solving
2.666666667 + x = 2.260776661

Solving for variable 'x'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + x = 2.260776661 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + x = 2.260776661 + -2.666666667
x = 2.260776661 + -2.666666667

Combine like terms: 2.260776661 + -2.666666667 = -0.405890006
x = -0.405890006

Simplifying
x = -0.405890006

Subproblem 2
x + 2.666666667 = -2.260776661

Simplifying
x + 2.666666667 = -2.260776661

Reorder the terms:
2.666666667 + x = -2.260776661

Solving
2.666666667 + x = -2.260776661

Solving for variable 'x'.

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

Add '-2.666666667' to each side of the equation.
2.666666667 + -2.666666667 + x = -2.260776661 + -2.666666667

Combine like terms: 2.666666667 + -2.666666667 = 0.000000000
0.000000000 + x = -2.260776661 + -2.666666667
x = -2.260776661 + -2.666666667

Combine like terms: -2.260776661 + -2.666666667 = -4.927443328
x = -4.927443328

Simplifying
x = -4.927443328

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.405890006, -4.927443328}