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

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

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

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

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

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

Combine like terms: 3x + -4x = -1x
(-6 + -1x + 2x2) + -1x(x + 1) = 0

Reorder the terms:
-6 + -1x + 2x2 + -1x(1 + x) = 0
-6 + -1x + 2x2 + (1 * -1x + x * -1x) = 0
-6 + -1x + 2x2 + (-1x + -1x2) = 0

Reorder the terms:
-6 + -1x + -1x + 2x2 + -1x2 = 0

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

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

Solving
-6 + -2x + 1x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -2x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + -2x + x2 = 0 + 6

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

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

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 = 6 + 1

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

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

Factor a perfect square on the left side:
(x + -1)(x + -1) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(x + -1) equal to 2.645751311 and -2.645751311.

Subproblem 1
x + -1 = 2.645751311

Simplifying
x + -1 = 2.645751311

Reorder the terms:
-1 + x = 2.645751311

Solving
-1 + x = 2.645751311

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 = 2.645751311 + 1

Combine like terms: -1 + 1 = 0
0 + x = 2.645751311 + 1
x = 2.645751311 + 1

Combine like terms: 2.645751311 + 1 = 3.645751311
x = 3.645751311

Simplifying
x = 3.645751311

Subproblem 2
x + -1 = -2.645751311

Simplifying
x + -1 = -2.645751311

Reorder the terms:
-1 + x = -2.645751311

Solving
-1 + x = -2.645751311

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 = -2.645751311 + 1

Combine like terms: -1 + 1 = 0
0 + x = -2.645751311 + 1
x = -2.645751311 + 1

Combine like terms: -2.645751311 + 1 = -1.645751311
x = -1.645751311

Simplifying
x = -1.645751311

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.645751311, -1.645751311}