4+6x(x-6)=10-8(x-3)

Solution for 4+6x(x-6)=10-8(x-3) equation:

Simplifying
4 + 6x(x + -6) = 10 + -8(x + -3)

Reorder the terms:
4 + 6x(-6 + x) = 10 + -8(x + -3)
4 + (-6 * 6x + x * 6x) = 10 + -8(x + -3)
4 + (-36x + 6x2) = 10 + -8(x + -3)

Reorder the terms:
4 + -36x + 6x2 = 10 + -8(-3 + x)
4 + -36x + 6x2 = 10 + (-3 * -8 + x * -8)
4 + -36x + 6x2 = 10 + (24 + -8x)

Combine like terms: 10 + 24 = 34
4 + -36x + 6x2 = 34 + -8x

Solving
4 + -36x + 6x2 = 34 + -8x

Solving for variable 'x'.

Reorder the terms:
4 + -34 + -36x + 8x + 6x2 = 34 + -8x + -34 + 8x

Combine like terms: 4 + -34 = -30
-30 + -36x + 8x + 6x2 = 34 + -8x + -34 + 8x

Combine like terms: -36x + 8x = -28x
-30 + -28x + 6x2 = 34 + -8x + -34 + 8x

Reorder the terms:
-30 + -28x + 6x2 = 34 + -34 + -8x + 8x

Combine like terms: 34 + -34 = 0
-30 + -28x + 6x2 = 0 + -8x + 8x
-30 + -28x + 6x2 = -8x + 8x

Combine like terms: -8x + 8x = 0
-30 + -28x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-15 + -14x + 3x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-15 + -14x + 3x2)' equal to zero and attempt to solve:

Simplifying
-15 + -14x + 3x2 = 0

Solving
-15 + -14x + 3x2 = 0

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

Divide each side by '3'.
-5 + -4.666666667x + x2 = 0

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -4.666666667x + 5 + x2 = 0 + 5

Reorder the terms:
-5 + 5 + -4.666666667x + x2 = 0 + 5

Combine like terms: -5 + 5 = 0
0 + -4.666666667x + x2 = 0 + 5
-4.666666667x + x2 = 0 + 5

Combine like terms: 0 + 5 = 5
-4.666666667x + x2 = 5

The x term is -4.666666667x.  Take half its coefficient (-2.333333334).
Square it (5.444444448) and add it to both sides.

Add '5.444444448' to each side of the equation.
-4.666666667x + 5.444444448 + x2 = 5 + 5.444444448

Reorder the terms:
5.444444448 + -4.666666667x + x2 = 5 + 5.444444448

Combine like terms: 5 + 5.444444448 = 10.444444448
5.444444448 + -4.666666667x + x2 = 10.444444448

Factor a perfect square on the left side:
(x + -2.333333334)(x + -2.333333334) = 10.444444448

Calculate the square root of the right side: 3.231786572

Break this problem into two subproblems by setting 
(x + -2.333333334) equal to 3.231786572 and -3.231786572.

Subproblem 1
x + -2.333333334 = 3.231786572

Simplifying
x + -2.333333334 = 3.231786572

Reorder the terms:
-2.333333334 + x = 3.231786572

Solving
-2.333333334 + x = 3.231786572

Solving for variable 'x'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + x = 3.231786572 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + x = 3.231786572 + 2.333333334
x = 3.231786572 + 2.333333334

Combine like terms: 3.231786572 + 2.333333334 = 5.565119906
x = 5.565119906

Simplifying
x = 5.565119906

Subproblem 2
x + -2.333333334 = -3.231786572

Simplifying
x + -2.333333334 = -3.231786572

Reorder the terms:
-2.333333334 + x = -3.231786572

Solving
-2.333333334 + x = -3.231786572

Solving for variable 'x'.

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

Add '2.333333334' to each side of the equation.
-2.333333334 + 2.333333334 + x = -3.231786572 + 2.333333334

Combine like terms: -2.333333334 + 2.333333334 = 0.000000000
0.000000000 + x = -3.231786572 + 2.333333334
x = -3.231786572 + 2.333333334

Combine like terms: -3.231786572 + 2.333333334 = -0.898453238
x = -0.898453238

Simplifying
x = -0.898453238

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {5.565119906, -0.898453238}
Solution
x = {5.565119906, -0.898453238}