4a^2+12a+6=0

Solution for 4a^2+12a+6=0 equation:

Simplifying
4a2 + 12a + 6 = 0

Reorder the terms:
6 + 12a + 4a2 = 0

Solving
6 + 12a + 4a2 = 0

Solving for variable 'a'.

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + 6a + 2a2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + 6a + 2a2)' equal to zero and attempt to solve:

Simplifying
3 + 6a + 2a2 = 0

Solving
3 + 6a + 2a2 = 0

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

Divide each side by '2'.
1.5 + 3a + a2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + 3a + -1.5 + a2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + 3a + a2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + 3a + a2 = 0 + -1.5
3a + a2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
3a + a2 = -1.5

The a term is 3a.  Take half its coefficient (1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
3a + 2.25 + a2 = -1.5 + 2.25

Reorder the terms:
2.25 + 3a + a2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + 3a + a2 = 0.75

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(a + 1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
a + 1.5 = 0.866025404

Simplifying
a + 1.5 = 0.866025404

Reorder the terms:
1.5 + a = 0.866025404

Solving
1.5 + a = 0.866025404

Solving for variable 'a'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + a = 0.866025404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = 0.866025404 + -1.5
a = 0.866025404 + -1.5

Combine like terms: 0.866025404 + -1.5 = -0.633974596
a = -0.633974596

Simplifying
a = -0.633974596

Subproblem 2
a + 1.5 = -0.866025404

Simplifying
a + 1.5 = -0.866025404

Reorder the terms:
1.5 + a = -0.866025404

Solving
1.5 + a = -0.866025404

Solving for variable 'a'.

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

Add '-1.5' to each side of the equation.
1.5 + -1.5 + a = -0.866025404 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = -0.866025404 + -1.5
a = -0.866025404 + -1.5

Combine like terms: -0.866025404 + -1.5 = -2.366025404
a = -2.366025404

Simplifying
a = -2.366025404

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.633974596, -2.366025404}
Solution
a = {-0.633974596, -2.366025404}