n+(n+2)(n+4)=186

Solution for n+(n+2)(n+4)=186 equation:

Simplifying
n + (n + 2)(n + 4) = 186

Reorder the terms:
n + (2 + n)(n + 4) = 186

Reorder the terms:
n + (2 + n)(4 + n) = 186

Multiply (2 + n) * (4 + n)
n + (2(4 + n) + n(4 + n)) = 186
n + ((4 * 2 + n * 2) + n(4 + n)) = 186
n + ((8 + 2n) + n(4 + n)) = 186
n + (8 + 2n + (4 * n + n * n)) = 186
n + (8 + 2n + (4n + n2)) = 186

Combine like terms: 2n + 4n = 6n
n + (8 + 6n + n2) = 186

Reorder the terms:
8 + n + 6n + n2 = 186

Combine like terms: n + 6n = 7n
8 + 7n + n2 = 186

Solving
8 + 7n + n2 = 186

Solving for variable 'n'.

Reorder the terms:
8 + -186 + 7n + n2 = 186 + -186

Combine like terms: 8 + -186 = -178
-178 + 7n + n2 = 186 + -186

Combine like terms: 186 + -186 = 0
-178 + 7n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '178' to each side of the equation.
-178 + 7n + 178 + n2 = 0 + 178

Reorder the terms:
-178 + 178 + 7n + n2 = 0 + 178

Combine like terms: -178 + 178 = 0
0 + 7n + n2 = 0 + 178
7n + n2 = 0 + 178

Combine like terms: 0 + 178 = 178
7n + n2 = 178

The n term is 7n.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7n + 12.25 + n2 = 178 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 178 + 12.25

Combine like terms: 178 + 12.25 = 190.25
12.25 + 7n + n2 = 190.25

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 190.25

Calculate the square root of the right side: 13.793114224

Break this problem into two subproblems by setting 
(n + 3.5) equal to 13.793114224 and -13.793114224.

Subproblem 1
n + 3.5 = 13.793114224

Simplifying
n + 3.5 = 13.793114224

Reorder the terms:
3.5 + n = 13.793114224

Solving
3.5 + n = 13.793114224

Solving for variable 'n'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + n = 13.793114224 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + n = 13.793114224 + -3.5
n = 13.793114224 + -3.5

Combine like terms: 13.793114224 + -3.5 = 10.293114224
n = 10.293114224

Simplifying
n = 10.293114224

Subproblem 2
n + 3.5 = -13.793114224

Simplifying
n + 3.5 = -13.793114224

Reorder the terms:
3.5 + n = -13.793114224

Solving
3.5 + n = -13.793114224

Solving for variable 'n'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + n = -13.793114224 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + n = -13.793114224 + -3.5
n = -13.793114224 + -3.5

Combine like terms: -13.793114224 + -3.5 = -17.293114224
n = -17.293114224

Simplifying
n = -17.293114224

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {10.293114224, -17.293114224}