s^4+2s^3+5s^2=0

Solution for s^4+2s^3+5s^2=0 equation:

Simplifying
s4 + 2s3 + 5s2 = 0

Reorder the terms:
5s2 + 2s3 + s4 = 0

Solving
5s2 + 2s3 + s4 = 0

Solving for variable 's'.

Factor out the Greatest Common Factor (GCF), 's2'.
s2(5 + 2s + s2) = 0

Subproblem 1
Set the factor 's2' equal to zero and attempt to solve:

Simplifying
s2 = 0

Solving
s2 = 0

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

Simplifying
s2 = 0

Take the square root of each side:
s = {0}
Subproblem 2
Set the factor '(5 + 2s + s2)' equal to zero and attempt to solve:

Simplifying
5 + 2s + s2 = 0

Solving
5 + 2s + s2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-5' to each side of the equation.
5 + 2s + -5 + s2 = 0 + -5

Reorder the terms:
5 + -5 + 2s + s2 = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 2s + s2 = 0 + -5
2s + s2 = 0 + -5

Combine like terms: 0 + -5 = -5
2s + s2 = -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 = -5 + 1

Reorder the terms:
1 + 2s + s2 = -5 + 1

Combine like terms: -5 + 1 = -4
1 + 2s + s2 = -4

Factor a perfect square on the left side:
(s + 1)(s + 1) = -4

Can't calculate square root of the right side.

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Solution
s = {0}