(x+1)(x+2)+(x+3)(x+4)-8=0

Solution for (x+1)(x+2)+(x+3)(x+4)-8=0 equation:

Simplifying
(x + 1)(x + 2) + (x + 3)(x + 4) + -8 = 0

Reorder the terms:
(1 + x)(x + 2) + (x + 3)(x + 4) + -8 = 0

Reorder the terms:
(1 + x)(2 + x) + (x + 3)(x + 4) + -8 = 0

Multiply (1 + x) * (2 + x)
(1(2 + x) + x(2 + x)) + (x + 3)(x + 4) + -8 = 0
((2 * 1 + x * 1) + x(2 + x)) + (x + 3)(x + 4) + -8 = 0
((2 + 1x) + x(2 + x)) + (x + 3)(x + 4) + -8 = 0
(2 + 1x + (2 * x + x * x)) + (x + 3)(x + 4) + -8 = 0
(2 + 1x + (2x + x2)) + (x + 3)(x + 4) + -8 = 0

Combine like terms: 1x + 2x = 3x
(2 + 3x + x2) + (x + 3)(x + 4) + -8 = 0

Reorder the terms:
2 + 3x + x2 + (3 + x)(x + 4) + -8 = 0

Reorder the terms:
2 + 3x + x2 + (3 + x)(4 + x) + -8 = 0

Multiply (3 + x) * (4 + x)
2 + 3x + x2 + (3(4 + x) + x(4 + x)) + -8 = 0
2 + 3x + x2 + ((4 * 3 + x * 3) + x(4 + x)) + -8 = 0
2 + 3x + x2 + ((12 + 3x) + x(4 + x)) + -8 = 0
2 + 3x + x2 + (12 + 3x + (4 * x + x * x)) + -8 = 0
2 + 3x + x2 + (12 + 3x + (4x + x2)) + -8 = 0

Combine like terms: 3x + 4x = 7x
2 + 3x + x2 + (12 + 7x + x2) + -8 = 0

Reorder the terms:
2 + 12 + -8 + 3x + 7x + x2 + x2 = 0

Combine like terms: 2 + 12 = 14
14 + -8 + 3x + 7x + x2 + x2 = 0

Combine like terms: 14 + -8 = 6
6 + 3x + 7x + x2 + x2 = 0

Combine like terms: 3x + 7x = 10x
6 + 10x + x2 + x2 = 0

Combine like terms: x2 + x2 = 2x2
6 + 10x + 2x2 = 0

Solving
6 + 10x + 2x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + 5x + x2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + 5x + x2)' equal to zero and attempt to solve:

Simplifying
3 + 5x + x2 = 0

Solving
3 + 5x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 5x + -3 + x2 = 0 + -3

Reorder the terms:
3 + -3 + 5x + x2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 5x + x2 = 0 + -3
5x + x2 = 0 + -3

Combine like terms: 0 + -3 = -3
5x + x2 = -3

The x term is 5x.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5x + 6.25 + x2 = -3 + 6.25

Reorder the terms:
6.25 + 5x + x2 = -3 + 6.25

Combine like terms: -3 + 6.25 = 3.25
6.25 + 5x + x2 = 3.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 3.25

Calculate the square root of the right side: 1.802775638

Break this problem into two subproblems by setting 
(x + 2.5) equal to 1.802775638 and -1.802775638.

Subproblem 1
x + 2.5 = 1.802775638

Simplifying
x + 2.5 = 1.802775638

Reorder the terms:
2.5 + x = 1.802775638

Solving
2.5 + x = 1.802775638

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = 1.802775638 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 1.802775638 + -2.5
x = 1.802775638 + -2.5

Combine like terms: 1.802775638 + -2.5 = -0.697224362
x = -0.697224362

Simplifying
x = -0.697224362

Subproblem 2
x + 2.5 = -1.802775638

Simplifying
x + 2.5 = -1.802775638

Reorder the terms:
2.5 + x = -1.802775638

Solving
2.5 + x = -1.802775638

Solving for variable 'x'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + x = -1.802775638 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -1.802775638 + -2.5
x = -1.802775638 + -2.5

Combine like terms: -1.802775638 + -2.5 = -4.302775638
x = -4.302775638

Simplifying
x = -4.302775638

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.697224362, -4.302775638}
Solution
x = {-0.697224362, -4.302775638}