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

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

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

Reorder the terms:
3(-6 + x) = 4x(x + 10)
(-6 * 3 + x * 3) = 4x(x + 10)
(-18 + 3x) = 4x(x + 10)

Reorder the terms:
-18 + 3x = 4x(10 + x)
-18 + 3x = (10 * 4x + x * 4x)
-18 + 3x = (40x + 4x2)

Solving
-18 + 3x = 40x + 4x2

Solving for variable 'x'.

Combine like terms: 3x + -40x = -37x
-18 + -37x + -4x2 = 40x + 4x2 + -40x + -4x2

Reorder the terms:
-18 + -37x + -4x2 = 40x + -40x + 4x2 + -4x2

Combine like terms: 40x + -40x = 0
-18 + -37x + -4x2 = 0 + 4x2 + -4x2
-18 + -37x + -4x2 = 4x2 + -4x2

Combine like terms: 4x2 + -4x2 = 0
-18 + -37x + -4x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(18 + 37x + 4x2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(18 + 37x + 4x2)' equal to zero and attempt to solve:

Simplifying
18 + 37x + 4x2 = 0

Solving
18 + 37x + 4x2 = 0

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

Divide each side by '4'.
4.5 + 9.25x + x2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + 9.25x + -4.5 + x2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + 9.25x + x2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + 9.25x + x2 = 0 + -4.5
9.25x + x2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
9.25x + x2 = -4.5

The x term is 9.25x.  Take half its coefficient (4.625).
Square it (21.390625) and add it to both sides.

Add '21.390625' to each side of the equation.
9.25x + 21.390625 + x2 = -4.5 + 21.390625

Reorder the terms:
21.390625 + 9.25x + x2 = -4.5 + 21.390625

Combine like terms: -4.5 + 21.390625 = 16.890625
21.390625 + 9.25x + x2 = 16.890625

Factor a perfect square on the left side:
(x + 4.625)(x + 4.625) = 16.890625

Calculate the square root of the right side: 4.109820556

Break this problem into two subproblems by setting 
(x + 4.625) equal to 4.109820556 and -4.109820556.

Subproblem 1
x + 4.625 = 4.109820556

Simplifying
x + 4.625 = 4.109820556

Reorder the terms:
4.625 + x = 4.109820556

Solving
4.625 + x = 4.109820556

Solving for variable 'x'.

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

Add '-4.625' to each side of the equation.
4.625 + -4.625 + x = 4.109820556 + -4.625

Combine like terms: 4.625 + -4.625 = 0.000
0.000 + x = 4.109820556 + -4.625
x = 4.109820556 + -4.625

Combine like terms: 4.109820556 + -4.625 = -0.515179444
x = -0.515179444

Simplifying
x = -0.515179444

Subproblem 2
x + 4.625 = -4.109820556

Simplifying
x + 4.625 = -4.109820556

Reorder the terms:
4.625 + x = -4.109820556

Solving
4.625 + x = -4.109820556

Solving for variable 'x'.

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

Add '-4.625' to each side of the equation.
4.625 + -4.625 + x = -4.109820556 + -4.625

Combine like terms: 4.625 + -4.625 = 0.000
0.000 + x = -4.109820556 + -4.625
x = -4.109820556 + -4.625

Combine like terms: -4.109820556 + -4.625 = -8.734820556
x = -8.734820556

Simplifying
x = -8.734820556

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.515179444, -8.734820556}
Solution
x = {-0.515179444, -8.734820556}