(2x+3)(x+4)=1

Solution for (2x+3)(x+4)=1 equation:

Simplifying
(2x + 3)(x + 4) = 1

Reorder the terms:
(3 + 2x)(x + 4) = 1

Reorder the terms:
(3 + 2x)(4 + x) = 1

Multiply (3 + 2x) * (4 + x)
(3(4 + x) + 2x * (4 + x)) = 1
((4 * 3 + x * 3) + 2x * (4 + x)) = 1
((12 + 3x) + 2x * (4 + x)) = 1
(12 + 3x + (4 * 2x + x * 2x)) = 1
(12 + 3x + (8x + 2x2)) = 1

Combine like terms: 3x + 8x = 11x
(12 + 11x + 2x2) = 1

Solving
12 + 11x + 2x2 = 1

Solving for variable 'x'.

Reorder the terms:
12 + -1 + 11x + 2x2 = 1 + -1

Combine like terms: 12 + -1 = 11
11 + 11x + 2x2 = 1 + -1

Combine like terms: 1 + -1 = 0
11 + 11x + 2x2 = 0

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

Divide each side by '2'.
5.5 + 5.5x + x2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + 5.5x + -5.5 + x2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + 5.5x + x2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + 5.5x + x2 = 0 + -5.5
5.5x + x2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
5.5x + x2 = -5.5

The x term is 5.5x.  Take half its coefficient (2.75).
Square it (7.5625) and add it to both sides.

Add '7.5625' to each side of the equation.
5.5x + 7.5625 + x2 = -5.5 + 7.5625

Reorder the terms:
7.5625 + 5.5x + x2 = -5.5 + 7.5625

Combine like terms: -5.5 + 7.5625 = 2.0625
7.5625 + 5.5x + x2 = 2.0625

Factor a perfect square on the left side:
(x + 2.75)(x + 2.75) = 2.0625

Calculate the square root of the right side: 1.436140662

Break this problem into two subproblems by setting 
(x + 2.75) equal to 1.436140662 and -1.436140662.

Subproblem 1
x + 2.75 = 1.436140662

Simplifying
x + 2.75 = 1.436140662

Reorder the terms:
2.75 + x = 1.436140662

Solving
2.75 + x = 1.436140662

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = 1.436140662 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = 1.436140662 + -2.75
x = 1.436140662 + -2.75

Combine like terms: 1.436140662 + -2.75 = -1.313859338
x = -1.313859338

Simplifying
x = -1.313859338

Subproblem 2
x + 2.75 = -1.436140662

Simplifying
x + 2.75 = -1.436140662

Reorder the terms:
2.75 + x = -1.436140662

Solving
2.75 + x = -1.436140662

Solving for variable 'x'.

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

Add '-2.75' to each side of the equation.
2.75 + -2.75 + x = -1.436140662 + -2.75

Combine like terms: 2.75 + -2.75 = 0.00
0.00 + x = -1.436140662 + -2.75
x = -1.436140662 + -2.75

Combine like terms: -1.436140662 + -2.75 = -4.186140662
x = -4.186140662

Simplifying
x = -4.186140662

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-1.313859338, -4.186140662}