(2x+3)(2x-5)-(2x+1)2=0

Solution for (2x+3)(2x-5)-(2x+1)2=0 equation:

Simplifying
(2x + 3)(2x + -5) + -1(2x + 1) * 2 = 0

Reorder the terms:
(3 + 2x)(2x + -5) + -1(2x + 1) * 2 = 0

Reorder the terms:
(3 + 2x)(-5 + 2x) + -1(2x + 1) * 2 = 0

Multiply (3 + 2x) * (-5 + 2x)
(3(-5 + 2x) + 2x * (-5 + 2x)) + -1(2x + 1) * 2 = 0
((-5 * 3 + 2x * 3) + 2x * (-5 + 2x)) + -1(2x + 1) * 2 = 0
((-15 + 6x) + 2x * (-5 + 2x)) + -1(2x + 1) * 2 = 0
(-15 + 6x + (-5 * 2x + 2x * 2x)) + -1(2x + 1) * 2 = 0
(-15 + 6x + (-10x + 4x2)) + -1(2x + 1) * 2 = 0

Combine like terms: 6x + -10x = -4x
(-15 + -4x + 4x2) + -1(2x + 1) * 2 = 0

Reorder the terms:
-15 + -4x + 4x2 + -1(1 + 2x) * 2 = 0

Reorder the terms for easier multiplication:
-15 + -4x + 4x2 + -1 * 2(1 + 2x) = 0

Multiply -1 * 2
-15 + -4x + 4x2 + -2(1 + 2x) = 0
-15 + -4x + 4x2 + (1 * -2 + 2x * -2) = 0
-15 + -4x + 4x2 + (-2 + -4x) = 0

Reorder the terms:
-15 + -2 + -4x + -4x + 4x2 = 0

Combine like terms: -15 + -2 = -17
-17 + -4x + -4x + 4x2 = 0

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

Solving
-17 + -8x + 4x2 = 0

Solving for variable 'x'.

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

Divide each side by '4'.
-4.25 + -2x + x2 = 0

Move the constant term to the right:

Add '4.25' to each side of the equation.
-4.25 + -2x + 4.25 + x2 = 0 + 4.25

Reorder the terms:
-4.25 + 4.25 + -2x + x2 = 0 + 4.25

Combine like terms: -4.25 + 4.25 = 0.00
0.00 + -2x + x2 = 0 + 4.25
-2x + x2 = 0 + 4.25

Combine like terms: 0 + 4.25 = 4.25
-2x + x2 = 4.25

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 4.25 + 1

Reorder the terms:
1 + -2x + x2 = 4.25 + 1

Combine like terms: 4.25 + 1 = 5.25
1 + -2x + x2 = 5.25

Factor a perfect square on the left side:
(x + -1)(x + -1) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(x + -1) equal to 2.291287847 and -2.291287847.

Subproblem 1
x + -1 = 2.291287847

Simplifying
x + -1 = 2.291287847

Reorder the terms:
-1 + x = 2.291287847

Solving
-1 + x = 2.291287847

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = 2.291287847 + 1

Combine like terms: -1 + 1 = 0
0 + x = 2.291287847 + 1
x = 2.291287847 + 1

Combine like terms: 2.291287847 + 1 = 3.291287847
x = 3.291287847

Simplifying
x = 3.291287847

Subproblem 2
x + -1 = -2.291287847

Simplifying
x + -1 = -2.291287847

Reorder the terms:
-1 + x = -2.291287847

Solving
-1 + x = -2.291287847

Solving for variable 'x'.

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

Add '1' to each side of the equation.
-1 + 1 + x = -2.291287847 + 1

Combine like terms: -1 + 1 = 0
0 + x = -2.291287847 + 1
x = -2.291287847 + 1

Combine like terms: -2.291287847 + 1 = -1.291287847
x = -1.291287847

Simplifying
x = -1.291287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.291287847, -1.291287847}