4x^3+5x^2+6x=0

Solution for 4x^3+5x^2+6x=0 equation:

Simplifying
4x3 + 5x2 + 6x = 0

Reorder the terms:
6x + 5x2 + 4x3 = 0

Solving
6x + 5x2 + 4x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(6 + 5x + 4x2) = 0

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

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(6 + 5x + 4x2)' equal to zero and attempt to solve:

Simplifying
6 + 5x + 4x2 = 0

Solving
6 + 5x + 4x2 = 0

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

Divide each side by '4'.
1.5 + 1.25x + x2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 1.25x + -1.5 + x2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 1.25x + x2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 1.25x + x2 = 0 + -1.5
1.25x + x2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
1.25x + x2 = -1.5

The x term is 1.25x.  Take half its coefficient (0.625).
Square it (0.390625) and add it to both sides.

Add '0.390625' to each side of the equation.
1.25x + 0.390625 + x2 = -1.5 + 0.390625

Reorder the terms:
0.390625 + 1.25x + x2 = -1.5 + 0.390625

Combine like terms: -1.5 + 0.390625 = -1.109375
0.390625 + 1.25x + x2 = -1.109375

Factor a perfect square on the left side:
(x + 0.625)(x + 0.625) = -1.109375

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
x = {0}