2x(2+x)3=21

Solution for 2x(2+x)3=21 equation:

Simplifying
2x(2 + x) * 3 = 21

Reorder the terms for easier multiplication:
2 * 3x(2 + x) = 21

Multiply 2 * 3
6x(2 + x) = 21
(2 * 6x + x * 6x) = 21
(12x + 6x2) = 21

Solving
12x + 6x2 = 21

Solving for variable 'x'.

Reorder the terms:
-21 + 12x + 6x2 = 21 + -21

Combine like terms: 21 + -21 = 0
-21 + 12x + 6x2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-7 + 4x + 2x2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-7 + 4x + 2x2)' equal to zero and attempt to solve:

Simplifying
-7 + 4x + 2x2 = 0

Solving
-7 + 4x + 2x2 = 0

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

Divide each side by '2'.
-3.5 + 2x + x2 = 0

Move the constant term to the right:

Add '3.5' to each side of the equation.
-3.5 + 2x + 3.5 + x2 = 0 + 3.5

Reorder the terms:
-3.5 + 3.5 + 2x + x2 = 0 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + 2x + x2 = 0 + 3.5
2x + x2 = 0 + 3.5

Combine like terms: 0 + 3.5 = 3.5
2x + x2 = 3.5

The x term is 2x.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2x + 1 + x2 = 3.5 + 1

Reorder the terms:
1 + 2x + x2 = 3.5 + 1

Combine like terms: 3.5 + 1 = 4.5
1 + 2x + x2 = 4.5

Factor a perfect square on the left side:
(x + 1)(x + 1) = 4.5

Calculate the square root of the right side: 2.121320344

Break this problem into two subproblems by setting 
(x + 1) equal to 2.121320344 and -2.121320344.

Subproblem 1
x + 1 = 2.121320344

Simplifying
x + 1 = 2.121320344

Reorder the terms:
1 + x = 2.121320344

Solving
1 + x = 2.121320344

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = 2.121320344 + -1

Combine like terms: 1 + -1 = 0
0 + x = 2.121320344 + -1
x = 2.121320344 + -1

Combine like terms: 2.121320344 + -1 = 1.121320344
x = 1.121320344

Simplifying
x = 1.121320344

Subproblem 2
x + 1 = -2.121320344

Simplifying
x + 1 = -2.121320344

Reorder the terms:
1 + x = -2.121320344

Solving
1 + x = -2.121320344

Solving for variable 'x'.

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

Add '-1' to each side of the equation.
1 + -1 + x = -2.121320344 + -1

Combine like terms: 1 + -1 = 0
0 + x = -2.121320344 + -1
x = -2.121320344 + -1

Combine like terms: -2.121320344 + -1 = -3.121320344
x = -3.121320344

Simplifying
x = -3.121320344

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.121320344, -3.121320344}
Solution
x = {1.121320344, -3.121320344}