(a+2)(3a+3)=180

Solution for (a+2)(3a+3)=180 equation:

Simplifying
(a + 2)(3a + 3) = 180

Reorder the terms:
(2 + a)(3a + 3) = 180

Reorder the terms:
(2 + a)(3 + 3a) = 180

Multiply (2 + a) * (3 + 3a)
(2(3 + 3a) + a(3 + 3a)) = 180
((3 * 2 + 3a * 2) + a(3 + 3a)) = 180
((6 + 6a) + a(3 + 3a)) = 180
(6 + 6a + (3 * a + 3a * a)) = 180
(6 + 6a + (3a + 3a2)) = 180

Combine like terms: 6a + 3a = 9a
(6 + 9a + 3a2) = 180

Solving
6 + 9a + 3a2 = 180

Solving for variable 'a'.

Reorder the terms:
6 + -180 + 9a + 3a2 = 180 + -180

Combine like terms: 6 + -180 = -174
-174 + 9a + 3a2 = 180 + -180

Combine like terms: 180 + -180 = 0
-174 + 9a + 3a2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-58 + 3a + a2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-58 + 3a + a2)' equal to zero and attempt to solve:

Simplifying
-58 + 3a + a2 = 0

Solving
-58 + 3a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '58' to each side of the equation.
-58 + 3a + 58 + a2 = 0 + 58

Reorder the terms:
-58 + 58 + 3a + a2 = 0 + 58

Combine like terms: -58 + 58 = 0
0 + 3a + a2 = 0 + 58
3a + a2 = 0 + 58

Combine like terms: 0 + 58 = 58
3a + a2 = 58

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 = 58 + 2.25

Reorder the terms:
2.25 + 3a + a2 = 58 + 2.25

Combine like terms: 58 + 2.25 = 60.25
2.25 + 3a + a2 = 60.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 60.25

Calculate the square root of the right side: 7.762087348

Break this problem into two subproblems by setting 
(a + 1.5) equal to 7.762087348 and -7.762087348.

Subproblem 1
a + 1.5 = 7.762087348

Simplifying
a + 1.5 = 7.762087348

Reorder the terms:
1.5 + a = 7.762087348

Solving
1.5 + a = 7.762087348

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 = 7.762087348 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = 7.762087348 + -1.5
a = 7.762087348 + -1.5

Combine like terms: 7.762087348 + -1.5 = 6.262087348
a = 6.262087348

Simplifying
a = 6.262087348

Subproblem 2
a + 1.5 = -7.762087348

Simplifying
a + 1.5 = -7.762087348

Reorder the terms:
1.5 + a = -7.762087348

Solving
1.5 + a = -7.762087348

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 = -7.762087348 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = -7.762087348 + -1.5
a = -7.762087348 + -1.5

Combine like terms: -7.762087348 + -1.5 = -9.262087348
a = -9.262087348

Simplifying
a = -9.262087348

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {6.262087348, -9.262087348}
Solution
a = {6.262087348, -9.262087348}