3a^3b^2+12a^2b^2+3ab^2=0

Solution for 3a^3b^2+12a^2b^2+3ab^2=0 equation:

Simplifying
3a3b2 + 12a2b2 + 3ab2 = 0

Reorder the terms:
3ab2 + 12a2b2 + 3a3b2 = 0

Solving
3ab2 + 12a2b2 + 3a3b2 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '3ab2'.
3ab2(1 + 4a + a2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor 'ab2' equal to zero and attempt to solve:

Simplifying
ab2 = 0

Solving
ab2 = 0

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

Simplifying
ab2 = 0

The solution to this equation could not be determined.

This subproblem is being ignored because a solution could not be determined.
Subproblem 2
Set the factor '(1 + 4a + a2)' equal to zero and attempt to solve:

Simplifying
1 + 4a + a2 = 0

Solving
1 + 4a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 4a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + 4a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 4a + a2 = 0 + -1
4a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
4a + a2 = -1

The a term is 4a.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4a + 4 + a2 = -1 + 4

Reorder the terms:
4 + 4a + a2 = -1 + 4

Combine like terms: -1 + 4 = 3
4 + 4a + a2 = 3

Factor a perfect square on the left side:
(a + 2)(a + 2) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(a + 2) equal to 1.732050808 and -1.732050808.

Subproblem 1
a + 2 = 1.732050808

Simplifying
a + 2 = 1.732050808

Reorder the terms:
2 + a = 1.732050808

Solving
2 + a = 1.732050808

Solving for variable 'a'.

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

Add '-2' to each side of the equation.
2 + -2 + a = 1.732050808 + -2

Combine like terms: 2 + -2 = 0
0 + a = 1.732050808 + -2
a = 1.732050808 + -2

Combine like terms: 1.732050808 + -2 = -0.267949192
a = -0.267949192

Simplifying
a = -0.267949192

Subproblem 2
a + 2 = -1.732050808

Simplifying
a + 2 = -1.732050808

Reorder the terms:
2 + a = -1.732050808

Solving
2 + a = -1.732050808

Solving for variable 'a'.

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

Add '-2' to each side of the equation.
2 + -2 + a = -1.732050808 + -2

Combine like terms: 2 + -2 = 0
0 + a = -1.732050808 + -2
a = -1.732050808 + -2

Combine like terms: -1.732050808 + -2 = -3.732050808
a = -3.732050808

Simplifying
a = -3.732050808

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.267949192, -3.732050808}
Solution
a = {-0.267949192, -3.732050808}