(x+1)(x+2)=(x+4)(x-1)10

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Solution for (x+1)(x+2)=(x+4)(x-1)10 equation: 


Simplifying
(x + 1)(x + 2) = (x + 4)(x + -1) * 10

Reorder the terms:
(1 + x)(x + 2) = (x + 4)(x + -1) * 10

Reorder the terms:
(1 + x)(2 + x) = (x + 4)(x + -1) * 10

Multiply (1 + x) * (2 + x)
(1(2 + x) + x(2 + x)) = (x + 4)(x + -1) * 10
((2 * 1 + x * 1) + x(2 + x)) = (x + 4)(x + -1) * 10
((2 + 1x) + x(2 + x)) = (x + 4)(x + -1) * 10
(2 + 1x + (2 * x + x * x)) = (x + 4)(x + -1) * 10
(2 + 1x + (2x + x2)) = (x + 4)(x + -1) * 10

Combine like terms: 1x + 2x = 3x
(2 + 3x + x2) = (x + 4)(x + -1) * 10

Reorder the terms:
2 + 3x + x2 = (4 + x)(x + -1) * 10

Reorder the terms:
2 + 3x + x2 = (4 + x)(-1 + x) * 10

Reorder the terms for easier multiplication:
2 + 3x + x2 = 10(4 + x)(-1 + x)

Multiply (4 + x) * (-1 + x)
2 + 3x + x2 = 10(4(-1 + x) + x(-1 + x))
2 + 3x + x2 = 10((-1 * 4 + x * 4) + x(-1 + x))
2 + 3x + x2 = 10((-4 + 4x) + x(-1 + x))
2 + 3x + x2 = 10(-4 + 4x + (-1 * x + x * x))
2 + 3x + x2 = 10(-4 + 4x + (-1x + x2))

Combine like terms: 4x + -1x = 3x
2 + 3x + x2 = 10(-4 + 3x + x2)
2 + 3x + x2 = (-4 * 10 + 3x * 10 + x2 * 10)
2 + 3x + x2 = (-40 + 30x + 10x2)

Solving
2 + 3x + x2 = -40 + 30x + 10x2

Solving for variable 'x'.

Reorder the terms:
2 + 40 + 3x + -30x + x2 + -10x2 = -40 + 30x + 10x2 + 40 + -30x + -10x2

Combine like terms: 2 + 40 = 42
42 + 3x + -30x + x2 + -10x2 = -40 + 30x + 10x2 + 40 + -30x + -10x2

Combine like terms: 3x + -30x = -27x
42 + -27x + x2 + -10x2 = -40 + 30x + 10x2 + 40 + -30x + -10x2

Combine like terms: x2 + -10x2 = -9x2
42 + -27x + -9x2 = -40 + 30x + 10x2 + 40 + -30x + -10x2

Reorder the terms:
42 + -27x + -9x2 = -40 + 40 + 30x + -30x + 10x2 + -10x2

Combine like terms: -40 + 40 = 0
42 + -27x + -9x2 = 0 + 30x + -30x + 10x2 + -10x2
42 + -27x + -9x2 = 30x + -30x + 10x2 + -10x2

Combine like terms: 30x + -30x = 0
42 + -27x + -9x2 = 0 + 10x2 + -10x2
42 + -27x + -9x2 = 10x2 + -10x2

Combine like terms: 10x2 + -10x2 = 0
42 + -27x + -9x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(14 + -9x + -3x2) = 0

Ignore the factor 3.

Subproblem 1
Set the factor '(14 + -9x + -3x2)' equal to zero and attempt to solve:

Simplifying
14 + -9x + -3x2 = 0

Solving
14 + -9x + -3x2 = 0

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

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

Move the constant term to the right:

Add '4.666666667' to each side of the equation.
-4.666666667 + 3x + 4.666666667 + x2 = 0 + 4.666666667

Reorder the terms:
-4.666666667 + 4.666666667 + 3x + x2 = 0 + 4.666666667

Combine like terms: -4.666666667 + 4.666666667 = 0.000000000
0.000000000 + 3x + x2 = 0 + 4.666666667
3x + x2 = 0 + 4.666666667

Combine like terms: 0 + 4.666666667 = 4.666666667
3x + x2 = 4.666666667

The x term is 3x.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3x + 2.25 + x2 = 4.666666667 + 2.25

Reorder the terms:
2.25 + 3x + x2 = 4.666666667 + 2.25

Combine like terms: 4.666666667 + 2.25 = 6.916666667
2.25 + 3x + x2 = 6.916666667

Factor a perfect square on the left side:
(x + 1.5)(x + 1.5) = 6.916666667

Calculate the square root of the right side: 2.62995564

Break this problem into two subproblems by setting 
(x + 1.5) equal to 2.62995564 and -2.62995564.


Subproblem 1
x + 1.5 = 2.62995564

Simplifying
x + 1.5 = 2.62995564

Reorder the terms:
1.5 + x = 2.62995564

Solving
1.5 + x = 2.62995564

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = 2.62995564 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = 2.62995564 + -1.5
x = 2.62995564 + -1.5

Combine like terms: 2.62995564 + -1.5 = 1.12995564
x = 1.12995564

Simplifying
x = 1.12995564


Subproblem 2
x + 1.5 = -2.62995564

Simplifying
x + 1.5 = -2.62995564

Reorder the terms:
1.5 + x = -2.62995564

Solving
1.5 + x = -2.62995564

Solving for variable 'x'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + x = -2.62995564 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + x = -2.62995564 + -1.5
x = -2.62995564 + -1.5

Combine like terms: -2.62995564 + -1.5 = -4.12995564
x = -4.12995564

Simplifying
x = -4.12995564


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.12995564, -4.12995564}
Solution
x = {1.12995564, -4.12995564}

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