(2x+3)(8x+1)=(4x+5)+2

Solution for (2x+3)(8x+1)=(4x+5)+2 equation:

Simplifying
(2x + 3)(8x + 1) = (4x + 5) + 2

Reorder the terms:
(3 + 2x)(8x + 1) = (4x + 5) + 2

Reorder the terms:
(3 + 2x)(1 + 8x) = (4x + 5) + 2

Multiply (3 + 2x) * (1 + 8x)
(3(1 + 8x) + 2x * (1 + 8x)) = (4x + 5) + 2
((1 * 3 + 8x * 3) + 2x * (1 + 8x)) = (4x + 5) + 2
((3 + 24x) + 2x * (1 + 8x)) = (4x + 5) + 2
(3 + 24x + (1 * 2x + 8x * 2x)) = (4x + 5) + 2
(3 + 24x + (2x + 16x2)) = (4x + 5) + 2

Combine like terms: 24x + 2x = 26x
(3 + 26x + 16x2) = (4x + 5) + 2

Reorder the terms:
3 + 26x + 16x2 = (5 + 4x) + 2

Remove parenthesis around (5 + 4x)
3 + 26x + 16x2 = 5 + 4x + 2

Reorder the terms:
3 + 26x + 16x2 = 5 + 2 + 4x

Combine like terms: 5 + 2 = 7
3 + 26x + 16x2 = 7 + 4x

Solving
3 + 26x + 16x2 = 7 + 4x

Solving for variable 'x'.

Reorder the terms:
3 + -7 + 26x + -4x + 16x2 = 7 + 4x + -7 + -4x

Combine like terms: 3 + -7 = -4
-4 + 26x + -4x + 16x2 = 7 + 4x + -7 + -4x

Combine like terms: 26x + -4x = 22x
-4 + 22x + 16x2 = 7 + 4x + -7 + -4x

Reorder the terms:
-4 + 22x + 16x2 = 7 + -7 + 4x + -4x

Combine like terms: 7 + -7 = 0
-4 + 22x + 16x2 = 0 + 4x + -4x
-4 + 22x + 16x2 = 4x + -4x

Combine like terms: 4x + -4x = 0
-4 + 22x + 16x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-2 + 11x + 8x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-2 + 11x + 8x2)' equal to zero and attempt to solve:

Simplifying
-2 + 11x + 8x2 = 0

Solving
-2 + 11x + 8x2 = 0

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

Divide each side by '8'.
-0.25 + 1.375x + x2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + 1.375x + 0.25 + x2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + 1.375x + x2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + 1.375x + x2 = 0 + 0.25
1.375x + x2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
1.375x + x2 = 0.25

The x term is 1.375x.  Take half its coefficient (0.6875).
Square it (0.47265625) and add it to both sides.

Add '0.47265625' to each side of the equation.
1.375x + 0.47265625 + x2 = 0.25 + 0.47265625

Reorder the terms:
0.47265625 + 1.375x + x2 = 0.25 + 0.47265625

Combine like terms: 0.25 + 0.47265625 = 0.72265625
0.47265625 + 1.375x + x2 = 0.72265625

Factor a perfect square on the left side:
(x + 0.6875)(x + 0.6875) = 0.72265625

Calculate the square root of the right side: 0.850091907

Break this problem into two subproblems by setting 
(x + 0.6875) equal to 0.850091907 and -0.850091907.

Subproblem 1
x + 0.6875 = 0.850091907

Simplifying
x + 0.6875 = 0.850091907

Reorder the terms:
0.6875 + x = 0.850091907

Solving
0.6875 + x = 0.850091907

Solving for variable 'x'.

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

Add '-0.6875' to each side of the equation.
0.6875 + -0.6875 + x = 0.850091907 + -0.6875

Combine like terms: 0.6875 + -0.6875 = 0.0000
0.0000 + x = 0.850091907 + -0.6875
x = 0.850091907 + -0.6875

Combine like terms: 0.850091907 + -0.6875 = 0.162591907
x = 0.162591907

Simplifying
x = 0.162591907

Subproblem 2
x + 0.6875 = -0.850091907

Simplifying
x + 0.6875 = -0.850091907

Reorder the terms:
0.6875 + x = -0.850091907

Solving
0.6875 + x = -0.850091907

Solving for variable 'x'.

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

Add '-0.6875' to each side of the equation.
0.6875 + -0.6875 + x = -0.850091907 + -0.6875

Combine like terms: 0.6875 + -0.6875 = 0.0000
0.0000 + x = -0.850091907 + -0.6875
x = -0.850091907 + -0.6875

Combine like terms: -0.850091907 + -0.6875 = -1.537591907
x = -1.537591907

Simplifying
x = -1.537591907

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.162591907, -1.537591907}
Solution
x = {0.162591907, -1.537591907}