(4x+1)(2x-3)=3(x+4)

Solution for (4x+1)(2x-3)=3(x+4) equation:

Simplifying
(4x + 1)(2x + -3) = 3(x + 4)

Reorder the terms:
(1 + 4x)(2x + -3) = 3(x + 4)

Reorder the terms:
(1 + 4x)(-3 + 2x) = 3(x + 4)

Multiply (1 + 4x) * (-3 + 2x)
(1(-3 + 2x) + 4x * (-3 + 2x)) = 3(x + 4)
((-3 * 1 + 2x * 1) + 4x * (-3 + 2x)) = 3(x + 4)
((-3 + 2x) + 4x * (-3 + 2x)) = 3(x + 4)
(-3 + 2x + (-3 * 4x + 2x * 4x)) = 3(x + 4)
(-3 + 2x + (-12x + 8x2)) = 3(x + 4)

Combine like terms: 2x + -12x = -10x
(-3 + -10x + 8x2) = 3(x + 4)

Reorder the terms:
-3 + -10x + 8x2 = 3(4 + x)
-3 + -10x + 8x2 = (4 * 3 + x * 3)
-3 + -10x + 8x2 = (12 + 3x)

Solving
-3 + -10x + 8x2 = 12 + 3x

Solving for variable 'x'.

Reorder the terms:
-3 + -12 + -10x + -3x + 8x2 = 12 + 3x + -12 + -3x

Combine like terms: -3 + -12 = -15
-15 + -10x + -3x + 8x2 = 12 + 3x + -12 + -3x

Combine like terms: -10x + -3x = -13x
-15 + -13x + 8x2 = 12 + 3x + -12 + -3x

Reorder the terms:
-15 + -13x + 8x2 = 12 + -12 + 3x + -3x

Combine like terms: 12 + -12 = 0
-15 + -13x + 8x2 = 0 + 3x + -3x
-15 + -13x + 8x2 = 3x + -3x

Combine like terms: 3x + -3x = 0
-15 + -13x + 8x2 = 0

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

Divide each side by '8'.
-1.875 + -1.625x + x2 = 0

Move the constant term to the right:

Add '1.875' to each side of the equation.
-1.875 + -1.625x + 1.875 + x2 = 0 + 1.875

Reorder the terms:
-1.875 + 1.875 + -1.625x + x2 = 0 + 1.875

Combine like terms: -1.875 + 1.875 = 0.000
0.000 + -1.625x + x2 = 0 + 1.875
-1.625x + x2 = 0 + 1.875

Combine like terms: 0 + 1.875 = 1.875
-1.625x + x2 = 1.875

The x term is -1.625x.  Take half its coefficient (-0.8125).
Square it (0.66015625) and add it to both sides.

Add '0.66015625' to each side of the equation.
-1.625x + 0.66015625 + x2 = 1.875 + 0.66015625

Reorder the terms:
0.66015625 + -1.625x + x2 = 1.875 + 0.66015625

Combine like terms: 1.875 + 0.66015625 = 2.53515625
0.66015625 + -1.625x + x2 = 2.53515625

Factor a perfect square on the left side:
(x + -0.8125)(x + -0.8125) = 2.53515625

Calculate the square root of the right side: 1.5922174

Break this problem into two subproblems by setting 
(x + -0.8125) equal to 1.5922174 and -1.5922174.

Subproblem 1
x + -0.8125 = 1.5922174

Simplifying
x + -0.8125 = 1.5922174

Reorder the terms:
-0.8125 + x = 1.5922174

Solving
-0.8125 + x = 1.5922174

Solving for variable 'x'.

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

Add '0.8125' to each side of the equation.
-0.8125 + 0.8125 + x = 1.5922174 + 0.8125

Combine like terms: -0.8125 + 0.8125 = 0.0000
0.0000 + x = 1.5922174 + 0.8125
x = 1.5922174 + 0.8125

Combine like terms: 1.5922174 + 0.8125 = 2.4047174
x = 2.4047174

Simplifying
x = 2.4047174

Subproblem 2
x + -0.8125 = -1.5922174

Simplifying
x + -0.8125 = -1.5922174

Reorder the terms:
-0.8125 + x = -1.5922174

Solving
-0.8125 + x = -1.5922174

Solving for variable 'x'.

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

Add '0.8125' to each side of the equation.
-0.8125 + 0.8125 + x = -1.5922174 + 0.8125

Combine like terms: -0.8125 + 0.8125 = 0.0000
0.0000 + x = -1.5922174 + 0.8125
x = -1.5922174 + 0.8125

Combine like terms: -1.5922174 + 0.8125 = -0.7797174
x = -0.7797174

Simplifying
x = -0.7797174

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.4047174, -0.7797174}