x(x+1)+(x+2)(x+3)=42

Solution for x(x+1)+(x+2)(x+3)=42 equation:

Simplifying
x(x + 1) + (x + 2)(x + 3) = 42

Reorder the terms:
x(1 + x) + (x + 2)(x + 3) = 42
(1 * x + x * x) + (x + 2)(x + 3) = 42
(1x + x2) + (x + 2)(x + 3) = 42

Reorder the terms:
1x + x2 + (2 + x)(x + 3) = 42

Reorder the terms:
1x + x2 + (2 + x)(3 + x) = 42

Multiply (2 + x) * (3 + x)
1x + x2 + (2(3 + x) + x(3 + x)) = 42
1x + x2 + ((3 * 2 + x * 2) + x(3 + x)) = 42
1x + x2 + ((6 + 2x) + x(3 + x)) = 42
1x + x2 + (6 + 2x + (3 * x + x * x)) = 42
1x + x2 + (6 + 2x + (3x + x2)) = 42

Combine like terms: 2x + 3x = 5x
1x + x2 + (6 + 5x + x2) = 42

Reorder the terms:
6 + 1x + 5x + x2 + x2 = 42

Combine like terms: 1x + 5x = 6x
6 + 6x + x2 + x2 = 42

Combine like terms: x2 + x2 = 2x2
6 + 6x + 2x2 = 42

Solving
6 + 6x + 2x2 = 42

Solving for variable 'x'.

Reorder the terms:
6 + -42 + 6x + 2x2 = 42 + -42

Combine like terms: 6 + -42 = -36
-36 + 6x + 2x2 = 42 + -42

Combine like terms: 42 + -42 = 0
-36 + 6x + 2x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-18 + 3x + x2) = 0

Factor a trinomial.
2((-6 + -1x)(3 + -1x)) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-6 + -1x)' equal to zero and attempt to solve:

Simplifying
-6 + -1x = 0

Solving
-6 + -1x = 0

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

Add '6' to each side of the equation.
-6 + 6 + -1x = 0 + 6

Combine like terms: -6 + 6 = 0
0 + -1x = 0 + 6
-1x = 0 + 6

Combine like terms: 0 + 6 = 6
-1x = 6

Divide each side by '-1'.
x = -6

Simplifying
x = -6
Subproblem 2
Set the factor '(3 + -1x)' equal to zero and attempt to solve:

Simplifying
3 + -1x = 0

Solving
3 + -1x = 0

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

Add '-3' to each side of the equation.
3 + -3 + -1x = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1x = 0 + -3
-1x = 0 + -3

Combine like terms: 0 + -3 = -3
-1x = -3

Divide each side by '-1'.
x = 3

Simplifying
x = 3
Solution
x = {-6, 3}