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

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


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

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

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

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

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

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

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

Multiply (-1 + 2x) * (-5 + 4x)
1 + 3x + 2x2 = (-1(-5 + 4x) + 2x * (-5 + 4x))
1 + 3x + 2x2 = ((-5 * -1 + 4x * -1) + 2x * (-5 + 4x))
1 + 3x + 2x2 = ((5 + -4x) + 2x * (-5 + 4x))
1 + 3x + 2x2 = (5 + -4x + (-5 * 2x + 4x * 2x))
1 + 3x + 2x2 = (5 + -4x + (-10x + 8x2))

Combine like terms: -4x + -10x = -14x
1 + 3x + 2x2 = (5 + -14x + 8x2)

Solving
1 + 3x + 2x2 = 5 + -14x + 8x2

Solving for variable 'x'.

Reorder the terms:
1 + -5 + 3x + 14x + 2x2 + -8x2 = 5 + -14x + 8x2 + -5 + 14x + -8x2

Combine like terms: 1 + -5 = -4
-4 + 3x + 14x + 2x2 + -8x2 = 5 + -14x + 8x2 + -5 + 14x + -8x2

Combine like terms: 3x + 14x = 17x
-4 + 17x + 2x2 + -8x2 = 5 + -14x + 8x2 + -5 + 14x + -8x2

Combine like terms: 2x2 + -8x2 = -6x2
-4 + 17x + -6x2 = 5 + -14x + 8x2 + -5 + 14x + -8x2

Reorder the terms:
-4 + 17x + -6x2 = 5 + -5 + -14x + 14x + 8x2 + -8x2

Combine like terms: 5 + -5 = 0
-4 + 17x + -6x2 = 0 + -14x + 14x + 8x2 + -8x2
-4 + 17x + -6x2 = -14x + 14x + 8x2 + -8x2

Combine like terms: -14x + 14x = 0
-4 + 17x + -6x2 = 0 + 8x2 + -8x2
-4 + 17x + -6x2 = 8x2 + -8x2

Combine like terms: 8x2 + -8x2 = 0
-4 + 17x + -6x2 = 0

Begin completing the square.  Divide all terms by
-6 the coefficient of the squared term: 

Divide each side by '-6'.
0.6666666667 + -2.833333333x + x2 = 0

Move the constant term to the right:

Add '-0.6666666667' to each side of the equation.
0.6666666667 + -2.833333333x + -0.6666666667 + x2 = 0 + -0.6666666667

Reorder the terms:
0.6666666667 + -0.6666666667 + -2.833333333x + x2 = 0 + -0.6666666667

Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000
0.0000000000 + -2.833333333x + x2 = 0 + -0.6666666667
-2.833333333x + x2 = 0 + -0.6666666667

Combine like terms: 0 + -0.6666666667 = -0.6666666667
-2.833333333x + x2 = -0.6666666667

The x term is -2.833333333x.  Take half its coefficient (-1.416666667).
Square it (2.006944445) and add it to both sides.

Add '2.006944445' to each side of the equation.
-2.833333333x + 2.006944445 + x2 = -0.6666666667 + 2.006944445

Reorder the terms:
2.006944445 + -2.833333333x + x2 = -0.6666666667 + 2.006944445

Combine like terms: -0.6666666667 + 2.006944445 = 1.3402777783
2.006944445 + -2.833333333x + x2 = 1.3402777783

Factor a perfect square on the left side:
(x + -1.416666667)(x + -1.416666667) = 1.3402777783

Calculate the square root of the right side: 1.157703666

Break this problem into two subproblems by setting 
(x + -1.416666667) equal to 1.157703666 and -1.157703666.


Subproblem 1
x + -1.416666667 = 1.157703666

Simplifying
x + -1.416666667 = 1.157703666

Reorder the terms:
-1.416666667 + x = 1.157703666

Solving
-1.416666667 + x = 1.157703666

Solving for variable 'x'.

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

Add '1.416666667' to each side of the equation.
-1.416666667 + 1.416666667 + x = 1.157703666 + 1.416666667

Combine like terms: -1.416666667 + 1.416666667 = 0.000000000
0.000000000 + x = 1.157703666 + 1.416666667
x = 1.157703666 + 1.416666667

Combine like terms: 1.157703666 + 1.416666667 = 2.574370333
x = 2.574370333

Simplifying
x = 2.574370333


Subproblem 2
x + -1.416666667 = -1.157703666

Simplifying
x + -1.416666667 = -1.157703666

Reorder the terms:
-1.416666667 + x = -1.157703666

Solving
-1.416666667 + x = -1.157703666

Solving for variable 'x'.

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

Add '1.416666667' to each side of the equation.
-1.416666667 + 1.416666667 + x = -1.157703666 + 1.416666667

Combine like terms: -1.416666667 + 1.416666667 = 0.000000000
0.000000000 + x = -1.157703666 + 1.416666667
x = -1.157703666 + 1.416666667

Combine like terms: -1.157703666 + 1.416666667 = 0.258963001
x = 0.258963001

Simplifying
x = 0.258963001


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.574370333, 0.258963001}

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