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

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

Simplifying
(x + 2)(2x + -3) + 5x = (2x + 1)(x + 4) * 4

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

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

Multiply (2 + x) * (-3 + 2x)
(2(-3 + 2x) + x(-3 + 2x)) + 5x = (2x + 1)(x + 4) * 4
((-3 * 2 + 2x * 2) + x(-3 + 2x)) + 5x = (2x + 1)(x + 4) * 4
((-6 + 4x) + x(-3 + 2x)) + 5x = (2x + 1)(x + 4) * 4
(-6 + 4x + (-3 * x + 2x * x)) + 5x = (2x + 1)(x + 4) * 4
(-6 + 4x + (-3x + 2x2)) + 5x = (2x + 1)(x + 4) * 4

Combine like terms: 4x + -3x = 1x
(-6 + 1x + 2x2) + 5x = (2x + 1)(x + 4) * 4

Reorder the terms:
-6 + 1x + 5x + 2x2 = (2x + 1)(x + 4) * 4

Combine like terms: 1x + 5x = 6x
-6 + 6x + 2x2 = (2x + 1)(x + 4) * 4

Reorder the terms:
-6 + 6x + 2x2 = (1 + 2x)(x + 4) * 4

Reorder the terms:
-6 + 6x + 2x2 = (1 + 2x)(4 + x) * 4

Reorder the terms for easier multiplication:
-6 + 6x + 2x2 = 4(1 + 2x)(4 + x)

Multiply (1 + 2x) * (4 + x)
-6 + 6x + 2x2 = 4(1(4 + x) + 2x * (4 + x))
-6 + 6x + 2x2 = 4((4 * 1 + x * 1) + 2x * (4 + x))
-6 + 6x + 2x2 = 4((4 + 1x) + 2x * (4 + x))
-6 + 6x + 2x2 = 4(4 + 1x + (4 * 2x + x * 2x))
-6 + 6x + 2x2 = 4(4 + 1x + (8x + 2x2))

Combine like terms: 1x + 8x = 9x
-6 + 6x + 2x2 = 4(4 + 9x + 2x2)
-6 + 6x + 2x2 = (4 * 4 + 9x * 4 + 2x2 * 4)
-6 + 6x + 2x2 = (16 + 36x + 8x2)

Solving
-6 + 6x + 2x2 = 16 + 36x + 8x2

Solving for variable 'x'.

Reorder the terms:
-6 + -16 + 6x + -36x + 2x2 + -8x2 = 16 + 36x + 8x2 + -16 + -36x + -8x2

Combine like terms: -6 + -16 = -22
-22 + 6x + -36x + 2x2 + -8x2 = 16 + 36x + 8x2 + -16 + -36x + -8x2

Combine like terms: 6x + -36x = -30x
-22 + -30x + 2x2 + -8x2 = 16 + 36x + 8x2 + -16 + -36x + -8x2

Combine like terms: 2x2 + -8x2 = -6x2
-22 + -30x + -6x2 = 16 + 36x + 8x2 + -16 + -36x + -8x2

Reorder the terms:
-22 + -30x + -6x2 = 16 + -16 + 36x + -36x + 8x2 + -8x2

Combine like terms: 16 + -16 = 0
-22 + -30x + -6x2 = 0 + 36x + -36x + 8x2 + -8x2
-22 + -30x + -6x2 = 36x + -36x + 8x2 + -8x2

Combine like terms: 36x + -36x = 0
-22 + -30x + -6x2 = 0 + 8x2 + -8x2
-22 + -30x + -6x2 = 8x2 + -8x2

Combine like terms: 8x2 + -8x2 = 0
-22 + -30x + -6x2 = 0

Factor out the Greatest Common Factor (GCF), '-2'.
-2(11 + 15x + 3x2) = 0

Ignore the factor -2.
Subproblem 1
Set the factor '(11 + 15x + 3x2)' equal to zero and attempt to solve:

Simplifying
11 + 15x + 3x2 = 0

Solving
11 + 15x + 3x2 = 0

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

Divide each side by '3'.
3.666666667 + 5x + x2 = 0

Move the constant term to the right:

Add '-3.666666667' to each side of the equation.
3.666666667 + 5x + -3.666666667 + x2 = 0 + -3.666666667

Reorder the terms:
3.666666667 + -3.666666667 + 5x + x2 = 0 + -3.666666667

Combine like terms: 3.666666667 + -3.666666667 = 0.000000000
0.000000000 + 5x + x2 = 0 + -3.666666667
5x + x2 = 0 + -3.666666667

Combine like terms: 0 + -3.666666667 = -3.666666667
5x + x2 = -3.666666667

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = -3.666666667 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -3.666666667 + 6.25

Combine like terms: -3.666666667 + 6.25 = 2.583333333
6.25 + 5x + x2 = 2.583333333

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 2.583333333

Calculate the square root of the right side: 1.607275127

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1.607275127 and -1.607275127.

Subproblem 1
x + 2.5 = 1.607275127

Simplifying
x + 2.5 = 1.607275127

Reorder the terms:
2.5 + x = 1.607275127

Solving
2.5 + x = 1.607275127

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 1.607275127 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 1.607275127 + -2.5
x = 1.607275127 + -2.5

Combine like terms: 1.607275127 + -2.5 = -0.892724873
x = -0.892724873

Simplifying
x = -0.892724873

Subproblem 2
x + 2.5 = -1.607275127

Simplifying
x + 2.5 = -1.607275127

Reorder the terms:
2.5 + x = -1.607275127

Solving
2.5 + x = -1.607275127

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -1.607275127 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -1.607275127 + -2.5
x = -1.607275127 + -2.5

Combine like terms: -1.607275127 + -2.5 = -4.107275127
x = -4.107275127

Simplifying
x = -4.107275127

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.892724873, -4.107275127}
Solution
x = {-0.892724873, -4.107275127}