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

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


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

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

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

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

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

Reorder the terms:
6 + 5x + x2 + (1 + x)(x + 3) = (x + 1)(x + 2)

Reorder the terms:
6 + 5x + x2 + (1 + x)(3 + x) = (x + 1)(x + 2)

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

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

Reorder the terms:
6 + 3 + 5x + 4x + x2 + x2 = (x + 1)(x + 2)

Combine like terms: 6 + 3 = 9
9 + 5x + 4x + x2 + x2 = (x + 1)(x + 2)

Combine like terms: 5x + 4x = 9x
9 + 9x + x2 + x2 = (x + 1)(x + 2)

Combine like terms: x2 + x2 = 2x2
9 + 9x + 2x2 = (x + 1)(x + 2)

Reorder the terms:
9 + 9x + 2x2 = (1 + x)(x + 2)

Reorder the terms:
9 + 9x + 2x2 = (1 + x)(2 + x)

Multiply (1 + x) * (2 + x)
9 + 9x + 2x2 = (1(2 + x) + x(2 + x))
9 + 9x + 2x2 = ((2 * 1 + x * 1) + x(2 + x))
9 + 9x + 2x2 = ((2 + 1x) + x(2 + x))
9 + 9x + 2x2 = (2 + 1x + (2 * x + x * x))
9 + 9x + 2x2 = (2 + 1x + (2x + x2))

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

Solving
9 + 9x + 2x2 = 2 + 3x + x2

Solving for variable 'x'.

Reorder the terms:
9 + -2 + 9x + -3x + 2x2 + -1x2 = 2 + 3x + x2 + -2 + -3x + -1x2

Combine like terms: 9 + -2 = 7
7 + 9x + -3x + 2x2 + -1x2 = 2 + 3x + x2 + -2 + -3x + -1x2

Combine like terms: 9x + -3x = 6x
7 + 6x + 2x2 + -1x2 = 2 + 3x + x2 + -2 + -3x + -1x2

Combine like terms: 2x2 + -1x2 = 1x2
7 + 6x + 1x2 = 2 + 3x + x2 + -2 + -3x + -1x2

Reorder the terms:
7 + 6x + 1x2 = 2 + -2 + 3x + -3x + x2 + -1x2

Combine like terms: 2 + -2 = 0
7 + 6x + 1x2 = 0 + 3x + -3x + x2 + -1x2
7 + 6x + 1x2 = 3x + -3x + x2 + -1x2

Combine like terms: 3x + -3x = 0
7 + 6x + 1x2 = 0 + x2 + -1x2
7 + 6x + 1x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
7 + 6x + 1x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-7' to each side of the equation.
7 + 6x + -7 + x2 = 0 + -7

Reorder the terms:
7 + -7 + 6x + x2 = 0 + -7

Combine like terms: 7 + -7 = 0
0 + 6x + x2 = 0 + -7
6x + x2 = 0 + -7

Combine like terms: 0 + -7 = -7
6x + x2 = -7

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = -7 + 9

Reorder the terms:
9 + 6x + x2 = -7 + 9

Combine like terms: -7 + 9 = 2
9 + 6x + x2 = 2

Factor a perfect square on the left side:
(x + 3)(x + 3) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(x + 3) equal to 1.414213562 and -1.414213562.


Subproblem 1
x + 3 = 1.414213562

Simplifying
x + 3 = 1.414213562

Reorder the terms:
3 + x = 1.414213562

Solving
3 + x = 1.414213562

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = 1.414213562 + -3

Combine like terms: 3 + -3 = 0
0 + x = 1.414213562 + -3
x = 1.414213562 + -3

Combine like terms: 1.414213562 + -3 = -1.585786438
x = -1.585786438

Simplifying
x = -1.585786438


Subproblem 2
x + 3 = -1.414213562

Simplifying
x + 3 = -1.414213562

Reorder the terms:
3 + x = -1.414213562

Solving
3 + x = -1.414213562

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = -1.414213562 + -3

Combine like terms: 3 + -3 = 0
0 + x = -1.414213562 + -3
x = -1.414213562 + -3

Combine like terms: -1.414213562 + -3 = -4.414213562
x = -4.414213562

Simplifying
x = -4.414213562


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.585786438, -4.414213562}

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