s^2+4s+1=0

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

Simplifying
s2 + 4s + 1 = 0

Reorder the terms:
1 + 4s + s2 = 0

Solving
1 + 4s + s2 = 0

Solving for variable 's'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 4s + -1 + s2 = 0 + -1

Reorder the terms:
1 + -1 + 4s + s2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4s + s2 = 0 + -1
4s + s2 = 0 + -1

Combine like terms: 0 + -1 = -1
4s + s2 = -1

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

Add '4' to each side of the equation.
4s + 4 + s2 = -1 + 4

Reorder the terms:
4 + 4s + s2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + 4s + s2 = 3

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

Calculate the square root of the right side: 1.732050808

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

Subproblem 1
s + 2 = 1.732050808

Simplifying
s + 2 = 1.732050808

Reorder the terms:
2 + s = 1.732050808

Solving
2 + s = 1.732050808

Solving for variable 's'.

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

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

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

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

Simplifying
s = -0.267949192

Subproblem 2
s + 2 = -1.732050808

Simplifying
s + 2 = -1.732050808

Reorder the terms:
2 + s = -1.732050808

Solving
2 + s = -1.732050808

Solving for variable 's'.

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

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

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

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

Simplifying
s = -3.732050808

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