n=(4n+1)(2n+6)

Solution for n=(4n+1)(2n+6) equation:

Simplifying
n = (4n + 1)(2n + 6)

Reorder the terms:
n = (1 + 4n)(2n + 6)

Reorder the terms:
n = (1 + 4n)(6 + 2n)

Multiply (1 + 4n) * (6 + 2n)
n = (1(6 + 2n) + 4n * (6 + 2n))
n = ((6 * 1 + 2n * 1) + 4n * (6 + 2n))
n = ((6 + 2n) + 4n * (6 + 2n))
n = (6 + 2n + (6 * 4n + 2n * 4n))
n = (6 + 2n + (24n + 8n2))

Combine like terms: 2n + 24n = 26n
n = (6 + 26n + 8n2)

Solving
n = 6 + 26n + 8n2

Solving for variable 'n'.

Reorder the terms:
-6 + n + -26n + -8n2 = 6 + 26n + 8n2 + -6 + -26n + -8n2

Combine like terms: n + -26n = -25n
-6 + -25n + -8n2 = 6 + 26n + 8n2 + -6 + -26n + -8n2

Reorder the terms:
-6 + -25n + -8n2 = 6 + -6 + 26n + -26n + 8n2 + -8n2

Combine like terms: 6 + -6 = 0
-6 + -25n + -8n2 = 0 + 26n + -26n + 8n2 + -8n2
-6 + -25n + -8n2 = 26n + -26n + 8n2 + -8n2

Combine like terms: 26n + -26n = 0
-6 + -25n + -8n2 = 0 + 8n2 + -8n2
-6 + -25n + -8n2 = 8n2 + -8n2

Combine like terms: 8n2 + -8n2 = 0
-6 + -25n + -8n2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(6 + 25n + 8n2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(6 + 25n + 8n2)' equal to zero and attempt to solve:

Simplifying
6 + 25n + 8n2 = 0

Solving
6 + 25n + 8n2 = 0

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

Divide each side by '8'.
0.75 + 3.125n + n2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 3.125n + -0.75 + n2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 3.125n + n2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 3.125n + n2 = 0 + -0.75
3.125n + n2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
3.125n + n2 = -0.75

The n term is 3.125n.  Take half its coefficient (1.5625).
Square it (2.44140625) and add it to both sides.

Add '2.44140625' to each side of the equation.
3.125n + 2.44140625 + n2 = -0.75 + 2.44140625

Reorder the terms:
2.44140625 + 3.125n + n2 = -0.75 + 2.44140625

Combine like terms: -0.75 + 2.44140625 = 1.69140625
2.44140625 + 3.125n + n2 = 1.69140625

Factor a perfect square on the left side:
(n + 1.5625)(n + 1.5625) = 1.69140625

Calculate the square root of the right side: 1.300540753

Break this problem into two subproblems by setting 
(n + 1.5625) equal to 1.300540753 and -1.300540753.

Subproblem 1
n + 1.5625 = 1.300540753

Simplifying
n + 1.5625 = 1.300540753

Reorder the terms:
1.5625 + n = 1.300540753

Solving
1.5625 + n = 1.300540753

Solving for variable 'n'.

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

Add '-1.5625' to each side of the equation.
1.5625 + -1.5625 + n = 1.300540753 + -1.5625

Combine like terms: 1.5625 + -1.5625 = 0.0000
0.0000 + n = 1.300540753 + -1.5625
n = 1.300540753 + -1.5625

Combine like terms: 1.300540753 + -1.5625 = -0.261959247
n = -0.261959247

Simplifying
n = -0.261959247

Subproblem 2
n + 1.5625 = -1.300540753

Simplifying
n + 1.5625 = -1.300540753

Reorder the terms:
1.5625 + n = -1.300540753

Solving
1.5625 + n = -1.300540753

Solving for variable 'n'.

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

Add '-1.5625' to each side of the equation.
1.5625 + -1.5625 + n = -1.300540753 + -1.5625

Combine like terms: 1.5625 + -1.5625 = 0.0000
0.0000 + n = -1.300540753 + -1.5625
n = -1.300540753 + -1.5625

Combine like terms: -1.300540753 + -1.5625 = -2.863040753
n = -2.863040753

Simplifying
n = -2.863040753

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.261959247, -2.863040753}
Solution
n = {-0.261959247, -2.863040753}