(4x-3)(x+6)=(x+8)(2x+3)

Solution for (4x-3)(x+6)=(x+8)(2x+3) equation:

Simplifying
(4x + -3)(x + 6) = (x + 8)(2x + 3)

Reorder the terms:
(-3 + 4x)(x + 6) = (x + 8)(2x + 3)

Reorder the terms:
(-3 + 4x)(6 + x) = (x + 8)(2x + 3)

Multiply (-3 + 4x) * (6 + x)
(-3(6 + x) + 4x * (6 + x)) = (x + 8)(2x + 3)
((6 * -3 + x * -3) + 4x * (6 + x)) = (x + 8)(2x + 3)
((-18 + -3x) + 4x * (6 + x)) = (x + 8)(2x + 3)
(-18 + -3x + (6 * 4x + x * 4x)) = (x + 8)(2x + 3)
(-18 + -3x + (24x + 4x2)) = (x + 8)(2x + 3)

Combine like terms: -3x + 24x = 21x
(-18 + 21x + 4x2) = (x + 8)(2x + 3)

Reorder the terms:
-18 + 21x + 4x2 = (8 + x)(2x + 3)

Reorder the terms:
-18 + 21x + 4x2 = (8 + x)(3 + 2x)

Multiply (8 + x) * (3 + 2x)
-18 + 21x + 4x2 = (8(3 + 2x) + x(3 + 2x))
-18 + 21x + 4x2 = ((3 * 8 + 2x * 8) + x(3 + 2x))
-18 + 21x + 4x2 = ((24 + 16x) + x(3 + 2x))
-18 + 21x + 4x2 = (24 + 16x + (3 * x + 2x * x))
-18 + 21x + 4x2 = (24 + 16x + (3x + 2x2))

Combine like terms: 16x + 3x = 19x
-18 + 21x + 4x2 = (24 + 19x + 2x2)

Solving
-18 + 21x + 4x2 = 24 + 19x + 2x2

Solving for variable 'x'.

Reorder the terms:
-18 + -24 + 21x + -19x + 4x2 + -2x2 = 24 + 19x + 2x2 + -24 + -19x + -2x2

Combine like terms: -18 + -24 = -42
-42 + 21x + -19x + 4x2 + -2x2 = 24 + 19x + 2x2 + -24 + -19x + -2x2

Combine like terms: 21x + -19x = 2x
-42 + 2x + 4x2 + -2x2 = 24 + 19x + 2x2 + -24 + -19x + -2x2

Combine like terms: 4x2 + -2x2 = 2x2
-42 + 2x + 2x2 = 24 + 19x + 2x2 + -24 + -19x + -2x2

Reorder the terms:
-42 + 2x + 2x2 = 24 + -24 + 19x + -19x + 2x2 + -2x2

Combine like terms: 24 + -24 = 0
-42 + 2x + 2x2 = 0 + 19x + -19x + 2x2 + -2x2
-42 + 2x + 2x2 = 19x + -19x + 2x2 + -2x2

Combine like terms: 19x + -19x = 0
-42 + 2x + 2x2 = 0 + 2x2 + -2x2
-42 + 2x + 2x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
-42 + 2x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-21 + x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-21 + x + x2)' equal to zero and attempt to solve:

Simplifying
-21 + x + x2 = 0

Solving
-21 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + x + 21 + x2 = 0 + 21

Reorder the terms:
-21 + 21 + x + x2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + x + x2 = 0 + 21
x + x2 = 0 + 21

Combine like terms: 0 + 21 = 21
x + x2 = 21

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
x + 0.25 + x2 = 21 + 0.25

Reorder the terms:
0.25 + x + x2 = 21 + 0.25

Combine like terms: 21 + 0.25 = 21.25
0.25 + x + x2 = 21.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 21.25

Calculate the square root of the right side: 4.609772229

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.609772229 and -4.609772229.

Subproblem 1
x + 0.5 = 4.609772229

Simplifying
x + 0.5 = 4.609772229

Reorder the terms:
0.5 + x = 4.609772229

Solving
0.5 + x = 4.609772229

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.609772229 + -0.5
x = 4.609772229 + -0.5

Combine like terms: 4.609772229 + -0.5 = 4.109772229
x = 4.109772229

Simplifying
x = 4.109772229

Subproblem 2
x + 0.5 = -4.609772229

Simplifying
x + 0.5 = -4.609772229

Reorder the terms:
0.5 + x = -4.609772229

Solving
0.5 + x = -4.609772229

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -4.609772229 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.609772229 + -0.5
x = -4.609772229 + -0.5

Combine like terms: -4.609772229 + -0.5 = -5.109772229
x = -5.109772229

Simplifying
x = -5.109772229

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {4.109772229, -5.109772229}
Solution
x = {4.109772229, -5.109772229}