n^2+4n+1=0

Solution for n^2+4n+1=0 equation:

Simplifying
n2 + 4n + 1 = 0

Reorder the terms:
1 + 4n + n2 = 0

Solving
1 + 4n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 4n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + 4n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4n + n2 = 0 + -1
4n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
4n + n2 = -1

The n term is 4n.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4n + 4 + n2 = -1 + 4

Reorder the terms:
4 + 4n + n2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + 4n + n2 = 3

Factor a perfect square on the left side:
(n + 2)(n + 2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(n + 2) equal to 1.732050808 and -1.732050808.

Subproblem 1
n + 2 = 1.732050808

Simplifying
n + 2 = 1.732050808

Reorder the terms:
2 + n = 1.732050808

Solving
2 + n = 1.732050808

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = 1.732050808 + -2

Combine like terms: 2 + -2 = 0
0 + n = 1.732050808 + -2
n = 1.732050808 + -2

Combine like terms: 1.732050808 + -2 = -0.267949192
n = -0.267949192

Simplifying
n = -0.267949192

Subproblem 2
n + 2 = -1.732050808

Simplifying
n + 2 = -1.732050808

Reorder the terms:
2 + n = -1.732050808

Solving
2 + n = -1.732050808

Solving for variable 'n'.

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

Add '-2' to each side of the equation.
2 + -2 + n = -1.732050808 + -2

Combine like terms: 2 + -2 = 0
0 + n = -1.732050808 + -2
n = -1.732050808 + -2

Combine like terms: -1.732050808 + -2 = -3.732050808
n = -3.732050808

Simplifying
n = -3.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.267949192, -3.732050808}