w(4w+6)+2w=2(2w+w+7w-3)

Solution for w(4w+6)+2w=2(2w+w+7w-3) equation:

Simplifying
w(4w + 6) + 2w = 2(2w + w + 7w + -3)

Reorder the terms:
w(6 + 4w) + 2w = 2(2w + w + 7w + -3)
(6 * w + 4w * w) + 2w = 2(2w + w + 7w + -3)
(6w + 4w2) + 2w = 2(2w + w + 7w + -3)

Reorder the terms:
6w + 2w + 4w2 = 2(2w + w + 7w + -3)

Combine like terms: 6w + 2w = 8w
8w + 4w2 = 2(2w + w + 7w + -3)

Reorder the terms:
8w + 4w2 = 2(-3 + 2w + w + 7w)

Combine like terms: 2w + w = 3w
8w + 4w2 = 2(-3 + 3w + 7w)

Combine like terms: 3w + 7w = 10w
8w + 4w2 = 2(-3 + 10w)
8w + 4w2 = (-3 * 2 + 10w * 2)
8w + 4w2 = (-6 + 20w)

Solving
8w + 4w2 = -6 + 20w

Solving for variable 'w'.

Reorder the terms:
6 + 8w + -20w + 4w2 = -6 + 20w + 6 + -20w

Combine like terms: 8w + -20w = -12w
6 + -12w + 4w2 = -6 + 20w + 6 + -20w

Reorder the terms:
6 + -12w + 4w2 = -6 + 6 + 20w + -20w

Combine like terms: -6 + 6 = 0
6 + -12w + 4w2 = 0 + 20w + -20w
6 + -12w + 4w2 = 20w + -20w

Combine like terms: 20w + -20w = 0
6 + -12w + 4w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(3 + -6w + 2w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(3 + -6w + 2w2)' equal to zero and attempt to solve:

Simplifying
3 + -6w + 2w2 = 0

Solving
3 + -6w + 2w2 = 0

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

Divide each side by '2'.
1.5 + -3w + w2 = 0

Move the constant term to the right:

Add '-1.5' to each side of the equation.
1.5 + -3w + -1.5 + w2 = 0 + -1.5

Reorder the terms:
1.5 + -1.5 + -3w + w2 = 0 + -1.5

Combine like terms: 1.5 + -1.5 = 0.0
0.0 + -3w + w2 = 0 + -1.5
-3w + w2 = 0 + -1.5

Combine like terms: 0 + -1.5 = -1.5
-3w + w2 = -1.5

The w term is -3w.  Take half its coefficient (-1.5).
Square it (2.25) and add it to both sides.

Add '2.25' to each side of the equation.
-3w + 2.25 + w2 = -1.5 + 2.25

Reorder the terms:
2.25 + -3w + w2 = -1.5 + 2.25

Combine like terms: -1.5 + 2.25 = 0.75
2.25 + -3w + w2 = 0.75

Factor a perfect square on the left side:
(w + -1.5)(w + -1.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(w + -1.5) equal to 0.866025404 and -0.866025404.

Subproblem 1
w + -1.5 = 0.866025404

Simplifying
w + -1.5 = 0.866025404

Reorder the terms:
-1.5 + w = 0.866025404

Solving
-1.5 + w = 0.866025404

Solving for variable 'w'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + w = 0.866025404 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + w = 0.866025404 + 1.5
w = 0.866025404 + 1.5

Combine like terms: 0.866025404 + 1.5 = 2.366025404
w = 2.366025404

Simplifying
w = 2.366025404

Subproblem 2
w + -1.5 = -0.866025404

Simplifying
w + -1.5 = -0.866025404

Reorder the terms:
-1.5 + w = -0.866025404

Solving
-1.5 + w = -0.866025404

Solving for variable 'w'.

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

Add '1.5' to each side of the equation.
-1.5 + 1.5 + w = -0.866025404 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + w = -0.866025404 + 1.5
w = -0.866025404 + 1.5

Combine like terms: -0.866025404 + 1.5 = 0.633974596
w = 0.633974596

Simplifying
w = 0.633974596

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {2.366025404, 0.633974596}
Solution
w = {2.366025404, 0.633974596}