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

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


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

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

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

Multiply (5 + 4x) * (4 + x)
(5(4 + x) + 4x * (4 + x)) = (x + -3)(x + -2)
((4 * 5 + x * 5) + 4x * (4 + x)) = (x + -3)(x + -2)
((20 + 5x) + 4x * (4 + x)) = (x + -3)(x + -2)
(20 + 5x + (4 * 4x + x * 4x)) = (x + -3)(x + -2)
(20 + 5x + (16x + 4x2)) = (x + -3)(x + -2)

Combine like terms: 5x + 16x = 21x
(20 + 21x + 4x2) = (x + -3)(x + -2)

Reorder the terms:
20 + 21x + 4x2 = (-3 + x)(x + -2)

Reorder the terms:
20 + 21x + 4x2 = (-3 + x)(-2 + x)

Multiply (-3 + x) * (-2 + x)
20 + 21x + 4x2 = (-3(-2 + x) + x(-2 + x))
20 + 21x + 4x2 = ((-2 * -3 + x * -3) + x(-2 + x))
20 + 21x + 4x2 = ((6 + -3x) + x(-2 + x))
20 + 21x + 4x2 = (6 + -3x + (-2 * x + x * x))
20 + 21x + 4x2 = (6 + -3x + (-2x + x2))

Combine like terms: -3x + -2x = -5x
20 + 21x + 4x2 = (6 + -5x + x2)

Solving
20 + 21x + 4x2 = 6 + -5x + x2

Solving for variable 'x'.

Reorder the terms:
20 + -6 + 21x + 5x + 4x2 + -1x2 = 6 + -5x + x2 + -6 + 5x + -1x2

Combine like terms: 20 + -6 = 14
14 + 21x + 5x + 4x2 + -1x2 = 6 + -5x + x2 + -6 + 5x + -1x2

Combine like terms: 21x + 5x = 26x
14 + 26x + 4x2 + -1x2 = 6 + -5x + x2 + -6 + 5x + -1x2

Combine like terms: 4x2 + -1x2 = 3x2
14 + 26x + 3x2 = 6 + -5x + x2 + -6 + 5x + -1x2

Reorder the terms:
14 + 26x + 3x2 = 6 + -6 + -5x + 5x + x2 + -1x2

Combine like terms: 6 + -6 = 0
14 + 26x + 3x2 = 0 + -5x + 5x + x2 + -1x2
14 + 26x + 3x2 = -5x + 5x + x2 + -1x2

Combine like terms: -5x + 5x = 0
14 + 26x + 3x2 = 0 + x2 + -1x2
14 + 26x + 3x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
14 + 26x + 3x2 = 0

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

Divide each side by '3'.
4.666666667 + 8.666666667x + x2 = 0

Move the constant term to the right:

Add '-4.666666667' to each side of the equation.
4.666666667 + 8.666666667x + -4.666666667 + x2 = 0 + -4.666666667

Reorder the terms:
4.666666667 + -4.666666667 + 8.666666667x + x2 = 0 + -4.666666667

Combine like terms: 4.666666667 + -4.666666667 = 0.000000000
0.000000000 + 8.666666667x + x2 = 0 + -4.666666667
8.666666667x + x2 = 0 + -4.666666667

Combine like terms: 0 + -4.666666667 = -4.666666667
8.666666667x + x2 = -4.666666667

The x term is 8.666666667x.  Take half its coefficient (4.333333334).
Square it (18.77777778) and add it to both sides.

Add '18.77777778' to each side of the equation.
8.666666667x + 18.77777778 + x2 = -4.666666667 + 18.77777778

Reorder the terms:
18.77777778 + 8.666666667x + x2 = -4.666666667 + 18.77777778

Combine like terms: -4.666666667 + 18.77777778 = 14.111111113
18.77777778 + 8.666666667x + x2 = 14.111111113

Factor a perfect square on the left side:
(x + 4.333333334)(x + 4.333333334) = 14.111111113

Calculate the square root of the right side: 3.75647589

Break this problem into two subproblems by setting 
(x + 4.333333334) equal to 3.75647589 and -3.75647589.


Subproblem 1
x + 4.333333334 = 3.75647589

Simplifying
x + 4.333333334 = 3.75647589

Reorder the terms:
4.333333334 + x = 3.75647589

Solving
4.333333334 + x = 3.75647589

Solving for variable 'x'.

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

Add '-4.333333334' to each side of the equation.
4.333333334 + -4.333333334 + x = 3.75647589 + -4.333333334

Combine like terms: 4.333333334 + -4.333333334 = 0.000000000
0.000000000 + x = 3.75647589 + -4.333333334
x = 3.75647589 + -4.333333334

Combine like terms: 3.75647589 + -4.333333334 = -0.576857444
x = -0.576857444

Simplifying
x = -0.576857444


Subproblem 2
x + 4.333333334 = -3.75647589

Simplifying
x + 4.333333334 = -3.75647589

Reorder the terms:
4.333333334 + x = -3.75647589

Solving
4.333333334 + x = -3.75647589

Solving for variable 'x'.

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

Add '-4.333333334' to each side of the equation.
4.333333334 + -4.333333334 + x = -3.75647589 + -4.333333334

Combine like terms: 4.333333334 + -4.333333334 = 0.000000000
0.000000000 + x = -3.75647589 + -4.333333334
x = -3.75647589 + -4.333333334

Combine like terms: -3.75647589 + -4.333333334 = -8.089809224
x = -8.089809224

Simplifying
x = -8.089809224


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.576857444, -8.089809224}

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