3n(2n+4)=6

Solution for 3n(2n+4)=6 equation:

Simplifying
3n(2n + 4) = 6

Reorder the terms:
3n(4 + 2n) = 6
(4 * 3n + 2n * 3n) = 6
(12n + 6n2) = 6

Solving
12n + 6n2 = 6

Solving for variable 'n'.

Reorder the terms:
-6 + 12n + 6n2 = 6 + -6

Combine like terms: 6 + -6 = 0
-6 + 12n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '6'.
6(-1 + 2n + n2) = 0

Ignore the factor 6.
Subproblem 1
Set the factor '(-1 + 2n + n2)' equal to zero and attempt to solve:

Simplifying
-1 + 2n + n2 = 0

Solving
-1 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '1' to each side of the equation.
-1 + 2n + 1 + n2 = 0 + 1

Reorder the terms:
-1 + 1 + 2n + n2 = 0 + 1

Combine like terms: -1 + 1 = 0
0 + 2n + n2 = 0 + 1
2n + n2 = 0 + 1

Combine like terms: 0 + 1 = 1
2n + n2 = 1

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

Add '1' to each side of the equation.
2n + 1 + n2 = 1 + 1

Reorder the terms:
1 + 2n + n2 = 1 + 1

Combine like terms: 1 + 1 = 2
1 + 2n + n2 = 2

Factor a perfect square on the left side:
(n + 1)(n + 1) = 2

Calculate the square root of the right side: 1.414213562

Break this problem into two subproblems by setting 
(n + 1) equal to 1.414213562 and -1.414213562.

Subproblem 1
n + 1 = 1.414213562

Simplifying
n + 1 = 1.414213562

Reorder the terms:
1 + n = 1.414213562

Solving
1 + n = 1.414213562

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = 1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + n = 1.414213562 + -1
n = 1.414213562 + -1

Combine like terms: 1.414213562 + -1 = 0.414213562
n = 0.414213562

Simplifying
n = 0.414213562

Subproblem 2
n + 1 = -1.414213562

Simplifying
n + 1 = -1.414213562

Reorder the terms:
1 + n = -1.414213562

Solving
1 + n = -1.414213562

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = -1.414213562 + -1

Combine like terms: 1 + -1 = 0
0 + n = -1.414213562 + -1
n = -1.414213562 + -1

Combine like terms: -1.414213562 + -1 = -2.414213562
n = -2.414213562

Simplifying
n = -2.414213562

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.414213562, -2.414213562}
Solution
n = {0.414213562, -2.414213562}