(3a+2)+2(7a-6)(4a+9)=0

Solution for (3a+2)+2(7a-6)(4a+9)=0 equation:

Simplifying
(3a + 2) + 2(7a + -6)(4a + 9) = 0

Reorder the terms:
(2 + 3a) + 2(7a + -6)(4a + 9) = 0

Remove parenthesis around (2 + 3a)
2 + 3a + 2(7a + -6)(4a + 9) = 0

Reorder the terms:
2 + 3a + 2(-6 + 7a)(4a + 9) = 0

Reorder the terms:
2 + 3a + 2(-6 + 7a)(9 + 4a) = 0

Multiply (-6 + 7a) * (9 + 4a)
2 + 3a + 2(-6(9 + 4a) + 7a * (9 + 4a)) = 0
2 + 3a + 2((9 * -6 + 4a * -6) + 7a * (9 + 4a)) = 0
2 + 3a + 2((-54 + -24a) + 7a * (9 + 4a)) = 0
2 + 3a + 2(-54 + -24a + (9 * 7a + 4a * 7a)) = 0
2 + 3a + 2(-54 + -24a + (63a + 28a2)) = 0

Combine like terms: -24a + 63a = 39a
2 + 3a + 2(-54 + 39a + 28a2) = 0
2 + 3a + (-54 * 2 + 39a * 2 + 28a2 * 2) = 0
2 + 3a + (-108 + 78a + 56a2) = 0

Reorder the terms:
2 + -108 + 3a + 78a + 56a2 = 0

Combine like terms: 2 + -108 = -106
-106 + 3a + 78a + 56a2 = 0

Combine like terms: 3a + 78a = 81a
-106 + 81a + 56a2 = 0

Solving
-106 + 81a + 56a2 = 0

Solving for variable 'a'.

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

Divide each side by '56'.
-1.892857143 + 1.446428571a + a2 = 0

Move the constant term to the right:

Add '1.892857143' to each side of the equation.
-1.892857143 + 1.446428571a + 1.892857143 + a2 = 0 + 1.892857143

Reorder the terms:
-1.892857143 + 1.892857143 + 1.446428571a + a2 = 0 + 1.892857143

Combine like terms: -1.892857143 + 1.892857143 = 0.000000000
0.000000000 + 1.446428571a + a2 = 0 + 1.892857143
1.446428571a + a2 = 0 + 1.892857143

Combine like terms: 0 + 1.892857143 = 1.892857143
1.446428571a + a2 = 1.892857143

The a term is 1.446428571a.  Take half its coefficient (0.7232142855).
Square it (0.5230389028) and add it to both sides.

Add '0.5230389028' to each side of the equation.
1.446428571a + 0.5230389028 + a2 = 1.892857143 + 0.5230389028

Reorder the terms:
0.5230389028 + 1.446428571a + a2 = 1.892857143 + 0.5230389028

Combine like terms: 1.892857143 + 0.5230389028 = 2.4158960458
0.5230389028 + 1.446428571a + a2 = 2.4158960458

Factor a perfect square on the left side:
(a + 0.7232142855)(a + 0.7232142855) = 2.4158960458

Calculate the square root of the right side: 1.554315298

Break this problem into two subproblems by setting 
(a + 0.7232142855) equal to 1.554315298 and -1.554315298.

Subproblem 1
a + 0.7232142855 = 1.554315298

Simplifying
a + 0.7232142855 = 1.554315298

Reorder the terms:
0.7232142855 + a = 1.554315298

Solving
0.7232142855 + a = 1.554315298

Solving for variable 'a'.

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

Add '-0.7232142855' to each side of the equation.
0.7232142855 + -0.7232142855 + a = 1.554315298 + -0.7232142855

Combine like terms: 0.7232142855 + -0.7232142855 = 0.0000000000
0.0000000000 + a = 1.554315298 + -0.7232142855
a = 1.554315298 + -0.7232142855

Combine like terms: 1.554315298 + -0.7232142855 = 0.8311010125
a = 0.8311010125

Simplifying
a = 0.8311010125

Subproblem 2
a + 0.7232142855 = -1.554315298

Simplifying
a + 0.7232142855 = -1.554315298

Reorder the terms:
0.7232142855 + a = -1.554315298

Solving
0.7232142855 + a = -1.554315298

Solving for variable 'a'.

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

Add '-0.7232142855' to each side of the equation.
0.7232142855 + -0.7232142855 + a = -1.554315298 + -0.7232142855

Combine like terms: 0.7232142855 + -0.7232142855 = 0.0000000000
0.0000000000 + a = -1.554315298 + -0.7232142855
a = -1.554315298 + -0.7232142855

Combine like terms: -1.554315298 + -0.7232142855 = -2.2775295835
a = -2.2775295835

Simplifying
a = -2.2775295835

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {0.8311010125, -2.2775295835}