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

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

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

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

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

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

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

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

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

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

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

Reorder the terms:
1 + 2x + x2 = 4 + 4x + x2 + (3 + x)(x + 3)

Reorder the terms:
1 + 2x + x2 = 4 + 4x + x2 + (3 + x)(3 + x)

Multiply (3 + x) * (3 + x)
1 + 2x + x2 = 4 + 4x + x2 + (3(3 + x) + x(3 + x))
1 + 2x + x2 = 4 + 4x + x2 + ((3 * 3 + x * 3) + x(3 + x))
1 + 2x + x2 = 4 + 4x + x2 + ((9 + 3x) + x(3 + x))
1 + 2x + x2 = 4 + 4x + x2 + (9 + 3x + (3 * x + x * x))
1 + 2x + x2 = 4 + 4x + x2 + (9 + 3x + (3x + x2))

Combine like terms: 3x + 3x = 6x
1 + 2x + x2 = 4 + 4x + x2 + (9 + 6x + x2)

Reorder the terms:
1 + 2x + x2 = 4 + 9 + 4x + 6x + x2 + x2

Combine like terms: 4 + 9 = 13
1 + 2x + x2 = 13 + 4x + 6x + x2 + x2

Combine like terms: 4x + 6x = 10x
1 + 2x + x2 = 13 + 10x + x2 + x2

Combine like terms: x2 + x2 = 2x2
1 + 2x + x2 = 13 + 10x + 2x2

Solving
1 + 2x + x2 = 13 + 10x + 2x2

Solving for variable 'x'.

Reorder the terms:
1 + -13 + 2x + -10x + x2 + -2x2 = 13 + 10x + 2x2 + -13 + -10x + -2x2

Combine like terms: 1 + -13 = -12
-12 + 2x + -10x + x2 + -2x2 = 13 + 10x + 2x2 + -13 + -10x + -2x2

Combine like terms: 2x + -10x = -8x
-12 + -8x + x2 + -2x2 = 13 + 10x + 2x2 + -13 + -10x + -2x2

Combine like terms: x2 + -2x2 = -1x2
-12 + -8x + -1x2 = 13 + 10x + 2x2 + -13 + -10x + -2x2

Reorder the terms:
-12 + -8x + -1x2 = 13 + -13 + 10x + -10x + 2x2 + -2x2

Combine like terms: 13 + -13 = 0
-12 + -8x + -1x2 = 0 + 10x + -10x + 2x2 + -2x2
-12 + -8x + -1x2 = 10x + -10x + 2x2 + -2x2

Combine like terms: 10x + -10x = 0
-12 + -8x + -1x2 = 0 + 2x2 + -2x2
-12 + -8x + -1x2 = 2x2 + -2x2

Combine like terms: 2x2 + -2x2 = 0
-12 + -8x + -1x2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(12 + 8x + x2) = 0

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

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

Simplifying
6 + x = 0

Solving
6 + x = 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 + x = 0 + -6

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

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

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

Simplifying
2 + x = 0

Solving
2 + x = 0

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

Add '-2' to each side of the equation.
2 + -2 + x = 0 + -2

Combine like terms: 2 + -2 = 0
0 + x = 0 + -2
x = 0 + -2

Combine like terms: 0 + -2 = -2
x = -2

Simplifying
x = -2
Solution
x = {-6, -2}