(4x-8)(7x+21)=21

Solution for (4x-8)(7x+21)=21 equation:

Simplifying
(4x + -8)(7x + 21) = 21

Reorder the terms:
(-8 + 4x)(7x + 21) = 21

Reorder the terms:
(-8 + 4x)(21 + 7x) = 21

Multiply (-8 + 4x) * (21 + 7x)
(-8(21 + 7x) + 4x * (21 + 7x)) = 21
((21 * -8 + 7x * -8) + 4x * (21 + 7x)) = 21
((-168 + -56x) + 4x * (21 + 7x)) = 21
(-168 + -56x + (21 * 4x + 7x * 4x)) = 21
(-168 + -56x + (84x + 28x2)) = 21

Combine like terms: -56x + 84x = 28x
(-168 + 28x + 28x2) = 21

Solving
-168 + 28x + 28x2 = 21

Solving for variable 'x'.

Reorder the terms:
-168 + -21 + 28x + 28x2 = 21 + -21

Combine like terms: -168 + -21 = -189
-189 + 28x + 28x2 = 21 + -21

Combine like terms: 21 + -21 = 0
-189 + 28x + 28x2 = 0

Factor out the Greatest Common Factor (GCF), '7'.
7(-27 + 4x + 4x2) = 0

Ignore the factor 7.
Subproblem 1
Set the factor '(-27 + 4x + 4x2)' equal to zero and attempt to solve:

Simplifying
-27 + 4x + 4x2 = 0

Solving
-27 + 4x + 4x2 = 0

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

Divide each side by '4'.
-6.75 + x + x2 = 0

Move the constant term to the right:

Add '6.75' to each side of the equation.
-6.75 + x + 6.75 + x2 = 0 + 6.75

Reorder the terms:
-6.75 + 6.75 + x + x2 = 0 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x + x2 = 0 + 6.75
x + x2 = 0 + 6.75

Combine like terms: 0 + 6.75 = 6.75
x + x2 = 6.75

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.
 + 0.25 + x2 = 6.75 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 6.75 + 0.25

Combine like terms: 6.75 + 0.25 = 7
1.25 + x2 = 7

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 7

Calculate the square root of the right side: 2.645751311

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.645751311 and -2.645751311.

Subproblem 1
x + 0.5 = 2.645751311

Simplifying
x + 0.5 = 2.645751311

Reorder the terms:
0.5 + x = 2.645751311

Solving
0.5 + x = 2.645751311

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 = 2.645751311 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.645751311 + -0.5
x = 2.645751311 + -0.5

Combine like terms: 2.645751311 + -0.5 = 2.145751311
x = 2.145751311

Simplifying
x = 2.145751311

Subproblem 2
x + 0.5 = -2.645751311

Simplifying
x + 0.5 = -2.645751311

Reorder the terms:
0.5 + x = -2.645751311

Solving
0.5 + x = -2.645751311

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 = -2.645751311 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.645751311 + -0.5
x = -2.645751311 + -0.5

Combine like terms: -2.645751311 + -0.5 = -3.145751311
x = -3.145751311

Simplifying
x = -3.145751311

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.145751311, -3.145751311}
Solution
x = {2.145751311, -3.145751311}