4n+6=2n(n+3)

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

Simplifying
4n + 6 = 2n(n + 3)

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

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

Solving
6 + 4n = 6n + 2n2

Solving for variable 'n'.

Combine like terms: 4n + -6n = -2n
6 + -2n + -2n2 = 6n + 2n2 + -6n + -2n2

Reorder the terms:
6 + -2n + -2n2 = 6n + -6n + 2n2 + -2n2

Combine like terms: 6n + -6n = 0
6 + -2n + -2n2 = 0 + 2n2 + -2n2
6 + -2n + -2n2 = 2n2 + -2n2

Combine like terms: 2n2 + -2n2 = 0
6 + -2n + -2n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + -1n + -1n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + -1n + -1n2)' equal to zero and attempt to solve:

Simplifying
3 + -1n + -1n2 = 0

Solving
3 + -1n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-3 + n + n2 = 0

Move the constant term to the right:

Add '3' to each side of the equation.
-3 + n + 3 + n2 = 0 + 3

Reorder the terms:
-3 + 3 + n + n2 = 0 + 3

Combine like terms: -3 + 3 = 0
0 + n + n2 = 0 + 3
n + n2 = 0 + 3

Combine like terms: 0 + 3 = 3
n + n2 = 3

The n term is n.  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 + n2 = 3 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + n2 = 3 + 0.25

Combine like terms: 3 + 0.25 = 3.25
1.25 + n2 = 3.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(n + 0.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
n + 0.5 = 1.802775638

Simplifying
n + 0.5 = 1.802775638

Reorder the terms:
0.5 + n = 1.802775638

Solving
0.5 + n = 1.802775638

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 1.802775638 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 1.802775638 + -0.5
n = 1.802775638 + -0.5

Combine like terms: 1.802775638 + -0.5 = 1.302775638
n = 1.302775638

Simplifying
n = 1.302775638

Subproblem 2
n + 0.5 = -1.802775638

Simplifying
n + 0.5 = -1.802775638

Reorder the terms:
0.5 + n = -1.802775638

Solving
0.5 + n = -1.802775638

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -1.802775638 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -1.802775638 + -0.5
n = -1.802775638 + -0.5

Combine like terms: -1.802775638 + -0.5 = -2.302775638
n = -2.302775638

Simplifying
n = -2.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.302775638, -2.302775638}
Solution
n = {1.302775638, -2.302775638}