(x+1)(x+2)+3(x+1)=0

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

Simplifying
(x + 1)(x + 2) + 3(x + 1) = 0

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

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

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

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

Reorder the terms:
2 + 3x + x2 + 3(1 + x) = 0
2 + 3x + x2 + (1 * 3 + x * 3) = 0
2 + 3x + x2 + (3 + 3x) = 0

Reorder the terms:
2 + 3 + 3x + 3x + x2 = 0

Combine like terms: 2 + 3 = 5
5 + 3x + 3x + x2 = 0

Combine like terms: 3x + 3x = 6x
5 + 6x + x2 = 0

Solving
5 + 6x + x2 = 0

Solving for variable 'x'.

Factor a trinomial.
(5 + x)(1 + x) = 0

Subproblem 1
Set the factor '(5 + x)' equal to zero and attempt to solve:

Simplifying
5 + x = 0

Solving
5 + x = 0

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

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

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

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

Simplifying
x = -5
Subproblem 2
Set the factor '(1 + x)' equal to zero and attempt to solve:

Simplifying
1 + x = 0

Solving
1 + x = 0

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

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

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

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

Simplifying
x = -1
Solution
x = {-5, -1}