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

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

Simplifying
4a3 + 6a2 + 5a = 0

Reorder the terms:
5a + 6a2 + 4a3 = 0

Solving
5a + 6a2 + 4a3 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), 'a'.
a(5 + 6a + 4a2) = 0

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

Simplifying
a = 0

Solving
a = 0

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

Simplifying
a = 0
Subproblem 2
Set the factor '(5 + 6a + 4a2)' equal to zero and attempt to solve:

Simplifying
5 + 6a + 4a2 = 0

Solving
5 + 6a + 4a2 = 0

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

Divide each side by '4'.
1.25 + 1.5a + a2 = 0

Move the constant term to the right:

Add '-1.25' to each side of the equation.
1.25 + 1.5a + -1.25 + a2 = 0 + -1.25

Reorder the terms:
1.25 + -1.25 + 1.5a + a2 = 0 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + 1.5a + a2 = 0 + -1.25
1.5a + a2 = 0 + -1.25

Combine like terms: 0 + -1.25 = -1.25
1.5a + a2 = -1.25

The a term is 1.5a.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5a + 0.5625 + a2 = -1.25 + 0.5625

Reorder the terms:
0.5625 + 1.5a + a2 = -1.25 + 0.5625

Combine like terms: -1.25 + 0.5625 = -0.6875
0.5625 + 1.5a + a2 = -0.6875

Factor a perfect square on the left side:
(a + 0.75)(a + 0.75) = -0.6875

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
a = {0}