(4x+3)(4x+3)=5

Solution for (4x+3)(4x+3)=5 equation:

Simplifying
(4x + 3)(4x + 3) = 5

Reorder the terms:
(3 + 4x)(4x + 3) = 5

Reorder the terms:
(3 + 4x)(3 + 4x) = 5

Multiply (3 + 4x) * (3 + 4x)
(3(3 + 4x) + 4x * (3 + 4x)) = 5
((3 * 3 + 4x * 3) + 4x * (3 + 4x)) = 5
((9 + 12x) + 4x * (3 + 4x)) = 5
(9 + 12x + (3 * 4x + 4x * 4x)) = 5
(9 + 12x + (12x + 16x2)) = 5

Combine like terms: 12x + 12x = 24x
(9 + 24x + 16x2) = 5

Solving
9 + 24x + 16x2 = 5

Solving for variable 'x'.

Reorder the terms:
9 + -5 + 24x + 16x2 = 5 + -5

Combine like terms: 9 + -5 = 4
4 + 24x + 16x2 = 5 + -5

Combine like terms: 5 + -5 = 0
4 + 24x + 16x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(1 + 6x + 4x2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(1 + 6x + 4x2)' equal to zero and attempt to solve:

Simplifying
1 + 6x + 4x2 = 0

Solving
1 + 6x + 4x2 = 0

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

Divide each side by '4'.
0.25 + 1.5x + x2 = 0

Move the constant term to the right:

Add '-0.25' to each side of the equation.
0.25 + 1.5x + -0.25 + x2 = 0 + -0.25

Reorder the terms:
0.25 + -0.25 + 1.5x + x2 = 0 + -0.25

Combine like terms: 0.25 + -0.25 = 0.00
0.00 + 1.5x + x2 = 0 + -0.25
1.5x + x2 = 0 + -0.25

Combine like terms: 0 + -0.25 = -0.25
1.5x + x2 = -0.25

The x term is 1.5x.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5x + 0.5625 + x2 = -0.25 + 0.5625

Reorder the terms:
0.5625 + 1.5x + x2 = -0.25 + 0.5625

Combine like terms: -0.25 + 0.5625 = 0.3125
0.5625 + 1.5x + x2 = 0.3125

Factor a perfect square on the left side:
(x + 0.75)(x + 0.75) = 0.3125

Calculate the square root of the right side: 0.559016994

Break this problem into two subproblems by setting 
(x + 0.75) equal to 0.559016994 and -0.559016994.

Subproblem 1
x + 0.75 = 0.559016994

Simplifying
x + 0.75 = 0.559016994

Reorder the terms:
0.75 + x = 0.559016994

Solving
0.75 + x = 0.559016994

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = 0.559016994 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = 0.559016994 + -0.75
x = 0.559016994 + -0.75

Combine like terms: 0.559016994 + -0.75 = -0.190983006
x = -0.190983006

Simplifying
x = -0.190983006

Subproblem 2
x + 0.75 = -0.559016994

Simplifying
x + 0.75 = -0.559016994

Reorder the terms:
0.75 + x = -0.559016994

Solving
0.75 + x = -0.559016994

Solving for variable 'x'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + x = -0.559016994 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + x = -0.559016994 + -0.75
x = -0.559016994 + -0.75

Combine like terms: -0.559016994 + -0.75 = -1.309016994
x = -1.309016994

Simplifying
x = -1.309016994

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.190983006, -1.309016994}
Solution
x = {-0.190983006, -1.309016994}