10-2x(x+3)=8+3(2x+5)

Solution for 10-2x(x+3)=8+3(2x+5) equation:

Simplifying
10 + -2x(x + 3) = 8 + 3(2x + 5)

Reorder the terms:
10 + -2x(3 + x) = 8 + 3(2x + 5)
10 + (3 * -2x + x * -2x) = 8 + 3(2x + 5)
10 + (-6x + -2x2) = 8 + 3(2x + 5)

Reorder the terms:
10 + -6x + -2x2 = 8 + 3(5 + 2x)
10 + -6x + -2x2 = 8 + (5 * 3 + 2x * 3)
10 + -6x + -2x2 = 8 + (15 + 6x)

Combine like terms: 8 + 15 = 23
10 + -6x + -2x2 = 23 + 6x

Solving
10 + -6x + -2x2 = 23 + 6x

Solving for variable 'x'.

Reorder the terms:
10 + -23 + -6x + -6x + -2x2 = 23 + 6x + -23 + -6x

Combine like terms: 10 + -23 = -13
-13 + -6x + -6x + -2x2 = 23 + 6x + -23 + -6x

Combine like terms: -6x + -6x = -12x
-13 + -12x + -2x2 = 23 + 6x + -23 + -6x

Reorder the terms:
-13 + -12x + -2x2 = 23 + -23 + 6x + -6x

Combine like terms: 23 + -23 = 0
-13 + -12x + -2x2 = 0 + 6x + -6x
-13 + -12x + -2x2 = 6x + -6x

Combine like terms: 6x + -6x = 0
-13 + -12x + -2x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(13 + 12x + 2x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(13 + 12x + 2x2)' equal to zero and attempt to solve:

Simplifying
13 + 12x + 2x2 = 0

Solving
13 + 12x + 2x2 = 0

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

Divide each side by '2'.
6.5 + 6x + x2 = 0

Move the constant term to the right:

Add '-6.5' to each side of the equation.
6.5 + 6x + -6.5 + x2 = 0 + -6.5

Reorder the terms:
6.5 + -6.5 + 6x + x2 = 0 + -6.5

Combine like terms: 6.5 + -6.5 = 0.0
0.0 + 6x + x2 = 0 + -6.5
6x + x2 = 0 + -6.5

Combine like terms: 0 + -6.5 = -6.5
6x + x2 = -6.5

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = -6.5 + 9

Reorder the terms:
9 + 6x + x2 = -6.5 + 9

Combine like terms: -6.5 + 9 = 2.5
9 + 6x + x2 = 2.5

Factor a perfect square on the left side:
(x + 3)(x + 3) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(x + 3) equal to 1.58113883 and -1.58113883.

Subproblem 1
x + 3 = 1.58113883

Simplifying
x + 3 = 1.58113883

Reorder the terms:
3 + x = 1.58113883

Solving
3 + x = 1.58113883

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = 1.58113883 + -3

Combine like terms: 3 + -3 = 0
0 + x = 1.58113883 + -3
x = 1.58113883 + -3

Combine like terms: 1.58113883 + -3 = -1.41886117
x = -1.41886117

Simplifying
x = -1.41886117

Subproblem 2
x + 3 = -1.58113883

Simplifying
x + 3 = -1.58113883

Reorder the terms:
3 + x = -1.58113883

Solving
3 + x = -1.58113883

Solving for variable 'x'.

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

Add '-3' to each side of the equation.
3 + -3 + x = -1.58113883 + -3

Combine like terms: 3 + -3 = 0
0 + x = -1.58113883 + -3
x = -1.58113883 + -3

Combine like terms: -1.58113883 + -3 = -4.58113883
x = -4.58113883

Simplifying
x = -4.58113883

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.41886117, -4.58113883}
Solution
x = {-1.41886117, -4.58113883}