(2a+3)(2a+3)=5

Solution for (2a+3)(2a+3)=5 equation:

Simplifying
(2a + 3)(2a + 3) = 5

Reorder the terms:
(3 + 2a)(2a + 3) = 5

Reorder the terms:
(3 + 2a)(3 + 2a) = 5

Multiply (3 + 2a) * (3 + 2a)
(3(3 + 2a) + 2a * (3 + 2a)) = 5
((3 * 3 + 2a * 3) + 2a * (3 + 2a)) = 5
((9 + 6a) + 2a * (3 + 2a)) = 5
(9 + 6a + (3 * 2a + 2a * 2a)) = 5
(9 + 6a + (6a + 4a2)) = 5

Combine like terms: 6a + 6a = 12a
(9 + 12a + 4a2) = 5

Solving
9 + 12a + 4a2 = 5

Solving for variable 'a'.

Reorder the terms:
9 + -5 + 12a + 4a2 = 5 + -5

Combine like terms: 9 + -5 = 4
4 + 12a + 4a2 = 5 + -5

Combine like terms: 5 + -5 = 0
4 + 12a + 4a2 = 0

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

Ignore the factor 4.
Subproblem 1
Set the factor '(1 + 3a + a2)' equal to zero and attempt to solve:

Simplifying
1 + 3a + a2 = 0

Solving
1 + 3a + a2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 3a + -1 + a2 = 0 + -1

Reorder the terms:
1 + -1 + 3a + a2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 3a + a2 = 0 + -1
3a + a2 = 0 + -1

Combine like terms: 0 + -1 = -1
3a + a2 = -1

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 + 2.25

Reorder the terms:
2.25 + 3a + a2 = -1 + 2.25

Combine like terms: -1 + 2.25 = 1.25
2.25 + 3a + a2 = 1.25

Factor a perfect square on the left side:
(a + 1.5)(a + 1.5) = 1.25

Calculate the square root of the right side: 1.118033989

Break this problem into two subproblems by setting 
(a + 1.5) equal to 1.118033989 and -1.118033989.

Subproblem 1
a + 1.5 = 1.118033989

Simplifying
a + 1.5 = 1.118033989

Reorder the terms:
1.5 + a = 1.118033989

Solving
1.5 + a = 1.118033989

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 = 1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = 1.118033989 + -1.5
a = 1.118033989 + -1.5

Combine like terms: 1.118033989 + -1.5 = -0.381966011
a = -0.381966011

Simplifying
a = -0.381966011

Subproblem 2
a + 1.5 = -1.118033989

Simplifying
a + 1.5 = -1.118033989

Reorder the terms:
1.5 + a = -1.118033989

Solving
1.5 + a = -1.118033989

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 = -1.118033989 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + a = -1.118033989 + -1.5
a = -1.118033989 + -1.5

Combine like terms: -1.118033989 + -1.5 = -2.618033989
a = -2.618033989

Simplifying
a = -2.618033989

Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {-0.381966011, -2.618033989}
Solution
a = {-0.381966011, -2.618033989}