0=2z^3+z^2+4z

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

Simplifying
0 = 2z3 + z2 + 4z

Reorder the terms:
0 = 4z + z2 + 2z3

Solving
0 = 4z + z2 + 2z3

Solving for variable 'z'.
Remove the zero:
-4z + -1z2 + -2z3 = 4z + z2 + 2z3 + -4z + -1z2 + -2z3

Reorder the terms:
-4z + -1z2 + -2z3 = 4z + -4z + z2 + -1z2 + 2z3 + -2z3

Combine like terms: 4z + -4z = 0
-4z + -1z2 + -2z3 = 0 + z2 + -1z2 + 2z3 + -2z3
-4z + -1z2 + -2z3 = z2 + -1z2 + 2z3 + -2z3

Combine like terms: z2 + -1z2 = 0
-4z + -1z2 + -2z3 = 0 + 2z3 + -2z3
-4z + -1z2 + -2z3 = 2z3 + -2z3

Combine like terms: 2z3 + -2z3 = 0
-4z + -1z2 + -2z3 = 0

Factor out the Greatest Common Factor (GCF), '-1z'.
-1z(4 + z + 2z2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor 'z' equal to zero and attempt to solve:

Simplifying
z = 0

Solving
z = 0

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

Simplifying
z = 0
Subproblem 2
Set the factor '(4 + z + 2z2)' equal to zero and attempt to solve:

Simplifying
4 + z + 2z2 = 0

Solving
4 + z + 2z2 = 0

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

Divide each side by '2'.
2 + 0.5z + z2 = 0

Move the constant term to the right:

Add '-2' to each side of the equation.
2 + 0.5z + -2 + z2 = 0 + -2

Reorder the terms:
2 + -2 + 0.5z + z2 = 0 + -2

Combine like terms: 2 + -2 = 0
0 + 0.5z + z2 = 0 + -2
0.5z + z2 = 0 + -2

Combine like terms: 0 + -2 = -2
0.5z + z2 = -2

The z term is z.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.5z + 0.25 + z2 = -2 + 0.25

Reorder the terms:
0.25 + 0.5z + z2 = -2 + 0.25

Combine like terms: -2 + 0.25 = -1.75
0.25 + 0.5z + z2 = -1.75

Factor a perfect square on the left side:
(z + 0.5)(z + 0.5) = -1.75

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
z = {0}