110592=(w+2)w(2)

Solution for 110592=(w+2)w(2) equation:

Simplifying
110592 = (w + 2) * w(2)

Reorder the terms:
110592 = (2 + w) * w(2)

Reorder the terms for easier multiplication:
110592 = 2w(2 + w)
110592 = (2 * 2w + w * 2w)
110592 = (4w + 2w2)

Solving
110592 = 4w + 2w2

Solving for variable 'w'.

Reorder the terms:
110592 + -4w + -2w2 = 4w + -4w + 2w2 + -2w2

Combine like terms: 4w + -4w = 0
110592 + -4w + -2w2 = 0 + 2w2 + -2w2
110592 + -4w + -2w2 = 2w2 + -2w2

Combine like terms: 2w2 + -2w2 = 0
110592 + -4w + -2w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(55296 + -2w + -1w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(55296 + -2w + -1w2)' equal to zero and attempt to solve:

Simplifying
55296 + -2w + -1w2 = 0

Solving
55296 + -2w + -1w2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-55296 + 2w + w2 = 0

Move the constant term to the right:

Add '55296' to each side of the equation.
-55296 + 2w + 55296 + w2 = 0 + 55296

Reorder the terms:
-55296 + 55296 + 2w + w2 = 0 + 55296

Combine like terms: -55296 + 55296 = 0
0 + 2w + w2 = 0 + 55296
2w + w2 = 0 + 55296

Combine like terms: 0 + 55296 = 55296
2w + w2 = 55296

The w term is 2w.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2w + 1 + w2 = 55296 + 1

Reorder the terms:
1 + 2w + w2 = 55296 + 1

Combine like terms: 55296 + 1 = 55297
1 + 2w + w2 = 55297

Factor a perfect square on the left side:
(w + 1)(w + 1) = 55297

Calculate the square root of the right side: 235.153141591

Break this problem into two subproblems by setting 
(w + 1) equal to 235.153141591 and -235.153141591.

Subproblem 1
w + 1 = 235.153141591

Simplifying
w + 1 = 235.153141591

Reorder the terms:
1 + w = 235.153141591

Solving
1 + w = 235.153141591

Solving for variable 'w'.

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

Add '-1' to each side of the equation.
1 + -1 + w = 235.153141591 + -1

Combine like terms: 1 + -1 = 0
0 + w = 235.153141591 + -1
w = 235.153141591 + -1

Combine like terms: 235.153141591 + -1 = 234.153141591
w = 234.153141591

Simplifying
w = 234.153141591

Subproblem 2
w + 1 = -235.153141591

Simplifying
w + 1 = -235.153141591

Reorder the terms:
1 + w = -235.153141591

Solving
1 + w = -235.153141591

Solving for variable 'w'.

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

Add '-1' to each side of the equation.
1 + -1 + w = -235.153141591 + -1

Combine like terms: 1 + -1 = 0
0 + w = -235.153141591 + -1
w = -235.153141591 + -1

Combine like terms: -235.153141591 + -1 = -236.153141591
w = -236.153141591

Simplifying
w = -236.153141591

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {234.153141591, -236.153141591}
Solution
w = {234.153141591, -236.153141591}