(x+2)(x+3)+(x+4)(x+5)=24

Solution for (x+2)(x+3)+(x+4)(x+5)=24 equation:

Simplifying
(x + 2)(x + 3) + (x + 4)(x + 5) = 24

Reorder the terms:
(2 + x)(x + 3) + (x + 4)(x + 5) = 24

Reorder the terms:
(2 + x)(3 + x) + (x + 4)(x + 5) = 24

Multiply (2 + x) * (3 + x)
(2(3 + x) + x(3 + x)) + (x + 4)(x + 5) = 24
((3 * 2 + x * 2) + x(3 + x)) + (x + 4)(x + 5) = 24
((6 + 2x) + x(3 + x)) + (x + 4)(x + 5) = 24
(6 + 2x + (3 * x + x * x)) + (x + 4)(x + 5) = 24
(6 + 2x + (3x + x2)) + (x + 4)(x + 5) = 24

Combine like terms: 2x + 3x = 5x
(6 + 5x + x2) + (x + 4)(x + 5) = 24

Reorder the terms:
6 + 5x + x2 + (4 + x)(x + 5) = 24

Reorder the terms:
6 + 5x + x2 + (4 + x)(5 + x) = 24

Multiply (4 + x) * (5 + x)
6 + 5x + x2 + (4(5 + x) + x(5 + x)) = 24
6 + 5x + x2 + ((5 * 4 + x * 4) + x(5 + x)) = 24
6 + 5x + x2 + ((20 + 4x) + x(5 + x)) = 24
6 + 5x + x2 + (20 + 4x + (5 * x + x * x)) = 24
6 + 5x + x2 + (20 + 4x + (5x + x2)) = 24

Combine like terms: 4x + 5x = 9x
6 + 5x + x2 + (20 + 9x + x2) = 24

Reorder the terms:
6 + 20 + 5x + 9x + x2 + x2 = 24

Combine like terms: 6 + 20 = 26
26 + 5x + 9x + x2 + x2 = 24

Combine like terms: 5x + 9x = 14x
26 + 14x + x2 + x2 = 24

Combine like terms: x2 + x2 = 2x2
26 + 14x + 2x2 = 24

Solving
26 + 14x + 2x2 = 24

Solving for variable 'x'.

Reorder the terms:
26 + -24 + 14x + 2x2 = 24 + -24

Combine like terms: 26 + -24 = 2
2 + 14x + 2x2 = 24 + -24

Combine like terms: 24 + -24 = 0
2 + 14x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(1 + 7x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(1 + 7x + x2)' equal to zero and attempt to solve:

Simplifying
1 + 7x + x2 = 0

Solving
1 + 7x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 7x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 7x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 7x + x2 = 0 + -1
7x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
7x + x2 = -1

The x term is 7x.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7x + 12.25 + x2 = -1 + 12.25

Reorder the terms:
12.25 + 7x + x2 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + 7x + x2 = 11.25

Factor a perfect square on the left side:
(x + 3.5)(x + 3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(x + 3.5) equal to 3.354101966 and -3.354101966.

Subproblem 1
x + 3.5 = 3.354101966

Simplifying
x + 3.5 = 3.354101966

Reorder the terms:
3.5 + x = 3.354101966

Solving
3.5 + x = 3.354101966

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = 3.354101966 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = 3.354101966 + -3.5
x = 3.354101966 + -3.5

Combine like terms: 3.354101966 + -3.5 = -0.145898034
x = -0.145898034

Simplifying
x = -0.145898034

Subproblem 2
x + 3.5 = -3.354101966

Simplifying
x + 3.5 = -3.354101966

Reorder the terms:
3.5 + x = -3.354101966

Solving
3.5 + x = -3.354101966

Solving for variable 'x'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + x = -3.354101966 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + x = -3.354101966 + -3.5
x = -3.354101966 + -3.5

Combine like terms: -3.354101966 + -3.5 = -6.854101966
x = -6.854101966

Simplifying
x = -6.854101966

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.145898034, -6.854101966}
Solution
x = {-0.145898034, -6.854101966}