(2m+3)(4m+3)=3

Solution for (2m+3)(4m+3)=3 equation:

Simplifying
(2m + 3)(4m + 3) = 3

Reorder the terms:
(3 + 2m)(4m + 3) = 3

Reorder the terms:
(3 + 2m)(3 + 4m) = 3

Multiply (3 + 2m) * (3 + 4m)
(3(3 + 4m) + 2m * (3 + 4m)) = 3
((3 * 3 + 4m * 3) + 2m * (3 + 4m)) = 3
((9 + 12m) + 2m * (3 + 4m)) = 3
(9 + 12m + (3 * 2m + 4m * 2m)) = 3
(9 + 12m + (6m + 8m2)) = 3

Combine like terms: 12m + 6m = 18m
(9 + 18m + 8m2) = 3

Solving
9 + 18m + 8m2 = 3

Solving for variable 'm'.

Reorder the terms:
9 + -3 + 18m + 8m2 = 3 + -3

Combine like terms: 9 + -3 = 6
6 + 18m + 8m2 = 3 + -3

Combine like terms: 3 + -3 = 0
6 + 18m + 8m2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + 9m + 4m2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + 9m + 4m2)' equal to zero and attempt to solve:

Simplifying
3 + 9m + 4m2 = 0

Solving
3 + 9m + 4m2 = 0

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

Divide each side by '4'.
0.75 + 2.25m + m2 = 0

Move the constant term to the right:

Add '-0.75' to each side of the equation.
0.75 + 2.25m + -0.75 + m2 = 0 + -0.75

Reorder the terms:
0.75 + -0.75 + 2.25m + m2 = 0 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + 2.25m + m2 = 0 + -0.75
2.25m + m2 = 0 + -0.75

Combine like terms: 0 + -0.75 = -0.75
2.25m + m2 = -0.75

The m term is 2.25m.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25m + 1.265625 + m2 = -0.75 + 1.265625

Reorder the terms:
1.265625 + 2.25m + m2 = -0.75 + 1.265625

Combine like terms: -0.75 + 1.265625 = 0.515625
1.265625 + 2.25m + m2 = 0.515625

Factor a perfect square on the left side:
(m + 1.125)(m + 1.125) = 0.515625

Calculate the square root of the right side: 0.718070331

Break this problem into two subproblems by setting 
(m + 1.125) equal to 0.718070331 and -0.718070331.

Subproblem 1
m + 1.125 = 0.718070331

Simplifying
m + 1.125 = 0.718070331

Reorder the terms:
1.125 + m = 0.718070331

Solving
1.125 + m = 0.718070331

Solving for variable 'm'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + m = 0.718070331 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + m = 0.718070331 + -1.125
m = 0.718070331 + -1.125

Combine like terms: 0.718070331 + -1.125 = -0.406929669
m = -0.406929669

Simplifying
m = -0.406929669

Subproblem 2
m + 1.125 = -0.718070331

Simplifying
m + 1.125 = -0.718070331

Reorder the terms:
1.125 + m = -0.718070331

Solving
1.125 + m = -0.718070331

Solving for variable 'm'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + m = -0.718070331 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + m = -0.718070331 + -1.125
m = -0.718070331 + -1.125

Combine like terms: -0.718070331 + -1.125 = -1.843070331
m = -1.843070331

Simplifying
m = -1.843070331

Solution
The solution to the problem is based on the solutions
from the subproblems.
m = {-0.406929669, -1.843070331}
Solution
m = {-0.406929669, -1.843070331}