2s^3+4s^2+s=0

Solution for 2s^3+4s^2+s=0 equation:

Simplifying
2s3 + 4s2 + s = 0

Reorder the terms:
s + 4s2 + 2s3 = 0

Solving
s + 4s2 + 2s3 = 0

Solving for variable 's'.

Factor out the Greatest Common Factor (GCF), 's'.
s(1 + 4s + 2s2) = 0

Subproblem 1
Set the factor 's' equal to zero and attempt to solve:

Simplifying
s = 0

Solving
s = 0

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

Simplifying
s = 0
Subproblem 2
Set the factor '(1 + 4s + 2s2)' equal to zero and attempt to solve:

Simplifying
1 + 4s + 2s2 = 0

Solving
1 + 4s + 2s2 = 0

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

Divide each side by '2'.
0.5 + 2s + s2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2s + -0.5 + s2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2s + s2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2s + s2 = 0 + -0.5
2s + s2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2s + s2 = -0.5

The s term is 2s.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2s + 1 + s2 = -0.5 + 1

Reorder the terms:
1 + 2s + s2 = -0.5 + 1

Combine like terms: -0.5 + 1 = 0.5
1 + 2s + s2 = 0.5

Factor a perfect square on the left side:
(s + 1)(s + 1) = 0.5

Calculate the square root of the right side: 0.707106781

Break this problem into two subproblems by setting 
(s + 1) equal to 0.707106781 and -0.707106781.

Subproblem 1
s + 1 = 0.707106781

Simplifying
s + 1 = 0.707106781

Reorder the terms:
1 + s = 0.707106781

Solving
1 + s = 0.707106781

Solving for variable 's'.

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

Add '-1' to each side of the equation.
1 + -1 + s = 0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + s = 0.707106781 + -1
s = 0.707106781 + -1

Combine like terms: 0.707106781 + -1 = -0.292893219
s = -0.292893219

Simplifying
s = -0.292893219

Subproblem 2
s + 1 = -0.707106781

Simplifying
s + 1 = -0.707106781

Reorder the terms:
1 + s = -0.707106781

Solving
1 + s = -0.707106781

Solving for variable 's'.

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

Add '-1' to each side of the equation.
1 + -1 + s = -0.707106781 + -1

Combine like terms: 1 + -1 = 0
0 + s = -0.707106781 + -1
s = -0.707106781 + -1

Combine like terms: -0.707106781 + -1 = -1.707106781
s = -1.707106781

Simplifying
s = -1.707106781

Solution
The solution to the problem is based on the solutions
from the subproblems.
s = {-0.292893219, -1.707106781}
Solution
s = {0, -0.292893219, -1.707106781}