(3x)(4x+1)=27

Solution for (3x)(4x+1)=27 equation:

Simplifying
(3x)(4x + 1) = 27

Remove parenthesis around (3x)
3x(4x + 1) = 27

Reorder the terms:
3x(1 + 4x) = 27
(1 * 3x + 4x * 3x) = 27
(3x + 12x2) = 27

Solving
3x + 12x2 = 27

Solving for variable 'x'.

Reorder the terms:
-27 + 3x + 12x2 = 27 + -27

Combine like terms: 27 + -27 = 0
-27 + 3x + 12x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-9 + x + 4x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-9 + x + 4x2)' equal to zero and attempt to solve:

Simplifying
-9 + x + 4x2 = 0

Solving
-9 + x + 4x2 = 0

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

Divide each side by '4'.
-2.25 + 0.25x + x2 = 0

Move the constant term to the right:

Add '2.25' to each side of the equation.
-2.25 + 0.25x + 2.25 + x2 = 0 + 2.25

Reorder the terms:
-2.25 + 2.25 + 0.25x + x2 = 0 + 2.25

Combine like terms: -2.25 + 2.25 = 0.00
0.00 + 0.25x + x2 = 0 + 2.25
0.25x + x2 = 0 + 2.25

Combine like terms: 0 + 2.25 = 2.25
0.25x + x2 = 2.25

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.
0.25x + 0.25 + x2 = 2.25 + 0.25

Reorder the terms:
0.25 + 0.25x + x2 = 2.25 + 0.25

Combine like terms: 2.25 + 0.25 = 2.5
0.25 + 0.25x + x2 = 2.5

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 2.5

Calculate the square root of the right side: 1.58113883

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.58113883 and -1.58113883.

Subproblem 1
x + 0.5 = 1.58113883

Simplifying
x + 0.5 = 1.58113883

Reorder the terms:
0.5 + x = 1.58113883

Solving
0.5 + x = 1.58113883

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 = 1.58113883 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.58113883 + -0.5
x = 1.58113883 + -0.5

Combine like terms: 1.58113883 + -0.5 = 1.08113883
x = 1.08113883

Simplifying
x = 1.08113883

Subproblem 2
x + 0.5 = -1.58113883

Simplifying
x + 0.5 = -1.58113883

Reorder the terms:
0.5 + x = -1.58113883

Solving
0.5 + x = -1.58113883

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 = -1.58113883 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.58113883 + -0.5
x = -1.58113883 + -0.5

Combine like terms: -1.58113883 + -0.5 = -2.08113883
x = -2.08113883

Simplifying
x = -2.08113883

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.08113883, -2.08113883}
Solution
x = {1.08113883, -2.08113883}