(3X+4)(2X-6)=106

Solution for (3X+4)(2X-6)=106 equation:

Simplifying
(3X + 4)(2X + -6) = 106

Reorder the terms:
(4 + 3X)(2X + -6) = 106

Reorder the terms:
(4 + 3X)(-6 + 2X) = 106

Multiply (4 + 3X) * (-6 + 2X)
(4(-6 + 2X) + 3X * (-6 + 2X)) = 106
((-6 * 4 + 2X * 4) + 3X * (-6 + 2X)) = 106
((-24 + 8X) + 3X * (-6 + 2X)) = 106
(-24 + 8X + (-6 * 3X + 2X * 3X)) = 106
(-24 + 8X + (-18X + 6X2)) = 106

Combine like terms: 8X + -18X = -10X
(-24 + -10X + 6X2) = 106

Solving
-24 + -10X + 6X2 = 106

Solving for variable 'X'.

Reorder the terms:
-24 + -106 + -10X + 6X2 = 106 + -106

Combine like terms: -24 + -106 = -130
-130 + -10X + 6X2 = 106 + -106

Combine like terms: 106 + -106 = 0
-130 + -10X + 6X2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-65 + -5X + 3X2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-65 + -5X + 3X2)' equal to zero and attempt to solve:

Simplifying
-65 + -5X + 3X2 = 0

Solving
-65 + -5X + 3X2 = 0

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

Divide each side by '3'.
-21.66666667 + -1.666666667X + X2 = 0

Move the constant term to the right:

Add '21.66666667' to each side of the equation.
-21.66666667 + -1.666666667X + 21.66666667 + X2 = 0 + 21.66666667

Reorder the terms:
-21.66666667 + 21.66666667 + -1.666666667X + X2 = 0 + 21.66666667

Combine like terms: -21.66666667 + 21.66666667 = 0.00000000
0.00000000 + -1.666666667X + X2 = 0 + 21.66666667
-1.666666667X + X2 = 0 + 21.66666667

Combine like terms: 0 + 21.66666667 = 21.66666667
-1.666666667X + X2 = 21.66666667

The X term is -1.666666667X.  Take half its coefficient (-0.8333333335).
Square it (0.6944444447) and add it to both sides.

Add '0.6944444447' to each side of the equation.
-1.666666667X + 0.6944444447 + X2 = 21.66666667 + 0.6944444447

Reorder the terms:
0.6944444447 + -1.666666667X + X2 = 21.66666667 + 0.6944444447

Combine like terms: 21.66666667 + 0.6944444447 = 22.3611111147
0.6944444447 + -1.666666667X + X2 = 22.3611111147

Factor a perfect square on the left side:
(X + -0.8333333335)(X + -0.8333333335) = 22.3611111147

Calculate the square root of the right side: 4.728753653

Break this problem into two subproblems by setting 
(X + -0.8333333335) equal to 4.728753653 and -4.728753653.

Subproblem 1
X + -0.8333333335 = 4.728753653

Simplifying
X + -0.8333333335 = 4.728753653

Reorder the terms:
-0.8333333335 + X = 4.728753653

Solving
-0.8333333335 + X = 4.728753653

Solving for variable 'X'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + X = 4.728753653 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + X = 4.728753653 + 0.8333333335
X = 4.728753653 + 0.8333333335

Combine like terms: 4.728753653 + 0.8333333335 = 5.5620869865
X = 5.5620869865

Simplifying
X = 5.5620869865

Subproblem 2
X + -0.8333333335 = -4.728753653

Simplifying
X + -0.8333333335 = -4.728753653

Reorder the terms:
-0.8333333335 + X = -4.728753653

Solving
-0.8333333335 + X = -4.728753653

Solving for variable 'X'.

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

Add '0.8333333335' to each side of the equation.
-0.8333333335 + 0.8333333335 + X = -4.728753653 + 0.8333333335

Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000
0.0000000000 + X = -4.728753653 + 0.8333333335
X = -4.728753653 + 0.8333333335

Combine like terms: -4.728753653 + 0.8333333335 = -3.8954203195
X = -3.8954203195

Simplifying
X = -3.8954203195

Solution
The solution to the problem is based on the solutions
from the subproblems.
X = {5.5620869865, -3.8954203195}
Solution
X = {5.5620869865, -3.8954203195}