(2x-3)-(3x^2-2)=(x+5)(x-5)

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

Simplifying
(2x + -3) + -1(3x2 + -2) = (x + 5)(x + -5)

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

Remove parenthesis around (-3 + 2x)
-3 + 2x + -1(3x2 + -2) = (x + 5)(x + -5)

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

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

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

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

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

Multiply (5 + x) * (-5 + x)
-1 + 2x + -3x2 = (5(-5 + x) + x(-5 + x))
-1 + 2x + -3x2 = ((-5 * 5 + x * 5) + x(-5 + x))
-1 + 2x + -3x2 = ((-25 + 5x) + x(-5 + x))
-1 + 2x + -3x2 = (-25 + 5x + (-5 * x + x * x))
-1 + 2x + -3x2 = (-25 + 5x + (-5x + x2))

Combine like terms: 5x + -5x = 0
-1 + 2x + -3x2 = (-25 + 0 + x2)
-1 + 2x + -3x2 = (-25 + x2)

Solving
-1 + 2x + -3x2 = -25 + x2

Solving for variable 'x'.

Reorder the terms:
-1 + 25 + 2x + -3x2 + -1x2 = -25 + x2 + 25 + -1x2

Combine like terms: -1 + 25 = 24
24 + 2x + -3x2 + -1x2 = -25 + x2 + 25 + -1x2

Combine like terms: -3x2 + -1x2 = -4x2
24 + 2x + -4x2 = -25 + x2 + 25 + -1x2

Reorder the terms:
24 + 2x + -4x2 = -25 + 25 + x2 + -1x2

Combine like terms: -25 + 25 = 0
24 + 2x + -4x2 = 0 + x2 + -1x2
24 + 2x + -4x2 = x2 + -1x2

Combine like terms: x2 + -1x2 = 0
24 + 2x + -4x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(12 + x + -2x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(12 + x + -2x2)' equal to zero and attempt to solve:

Simplifying
12 + x + -2x2 = 0

Solving
12 + x + -2x2 = 0

Begin completing the square.  Divide all terms by
-2 the coefficient of the squared term: 

Divide each side by '-2'.
-6 + -0.5x + x2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + -0.5x + 6 + x2 = 0 + 6

Reorder the terms:
-6 + 6 + -0.5x + x2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -0.5x + x2 = 0 + 6
-0.5x + x2 = 0 + 6

Combine like terms: 0 + 6 = 6
-0.5x + x2 = 6

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-0.5x + 0.25 + x2 = 6 + 0.25

Reorder the terms:
0.25 + -0.5x + x2 = 6 + 0.25

Combine like terms: 6 + 0.25 = 6.25
0.25 + -0.5x + x2 = 6.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 6.25

Calculate the square root of the right side: 2.5

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.5 and -2.5.

Subproblem 1
x + 0.5 = 2.5

Simplifying
x + 0.5 = 2.5

Reorder the terms:
0.5 + x = 2.5

Solving
0.5 + x = 2.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 2.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.5 + -0.5
x = 2.5 + -0.5

Combine like terms: 2.5 + -0.5 = 2
x = 2

Simplifying
x = 2

Subproblem 2
x + 0.5 = -2.5

Simplifying
x + 0.5 = -2.5

Reorder the terms:
0.5 + x = -2.5

Solving
0.5 + x = -2.5

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -2.5 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.5 + -0.5
x = -2.5 + -0.5

Combine like terms: -2.5 + -0.5 = -3
x = -3

Simplifying
x = -3

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2, -3}
Solution
x = {2, -3}