2w+2(3+w)2(3+w)=46

Solution for 2w+2(3+w)2(3+w)=46 equation:

Simplifying
2w + 2(3 + w) * 2(3 + w) = 46

Reorder the terms for easier multiplication:
2w + 2 * 2(3 + w)(3 + w) = 46

Multiply 2 * 2
2w + 4(3 + w)(3 + w) = 46

Multiply (3 + w) * (3 + w)
2w + 4(3(3 + w) + w(3 + w)) = 46
2w + 4((3 * 3 + w * 3) + w(3 + w)) = 46
2w + 4((9 + 3w) + w(3 + w)) = 46
2w + 4(9 + 3w + (3 * w + w * w)) = 46
2w + 4(9 + 3w + (3w + w2)) = 46

Combine like terms: 3w + 3w = 6w
2w + 4(9 + 6w + w2) = 46
2w + (9 * 4 + 6w * 4 + w2 * 4) = 46
2w + (36 + 24w + 4w2) = 46

Reorder the terms:
36 + 2w + 24w + 4w2 = 46

Combine like terms: 2w + 24w = 26w
36 + 26w + 4w2 = 46

Solving
36 + 26w + 4w2 = 46

Solving for variable 'w'.

Reorder the terms:
36 + -46 + 26w + 4w2 = 46 + -46

Combine like terms: 36 + -46 = -10
-10 + 26w + 4w2 = 46 + -46

Combine like terms: 46 + -46 = 0
-10 + 26w + 4w2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-5 + 13w + 2w2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-5 + 13w + 2w2)' equal to zero and attempt to solve:

Simplifying
-5 + 13w + 2w2 = 0

Solving
-5 + 13w + 2w2 = 0

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

Divide each side by '2'.
-2.5 + 6.5w + w2 = 0

Move the constant term to the right:

Add '2.5' to each side of the equation.
-2.5 + 6.5w + 2.5 + w2 = 0 + 2.5

Reorder the terms:
-2.5 + 2.5 + 6.5w + w2 = 0 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + 6.5w + w2 = 0 + 2.5
6.5w + w2 = 0 + 2.5

Combine like terms: 0 + 2.5 = 2.5
6.5w + w2 = 2.5

The w term is 6.5w.  Take half its coefficient (3.25).
Square it (10.5625) and add it to both sides.

Add '10.5625' to each side of the equation.
6.5w + 10.5625 + w2 = 2.5 + 10.5625

Reorder the terms:
10.5625 + 6.5w + w2 = 2.5 + 10.5625

Combine like terms: 2.5 + 10.5625 = 13.0625
10.5625 + 6.5w + w2 = 13.0625

Factor a perfect square on the left side:
(w + 3.25)(w + 3.25) = 13.0625

Calculate the square root of the right side: 3.614208074

Break this problem into two subproblems by setting 
(w + 3.25) equal to 3.614208074 and -3.614208074.

Subproblem 1
w + 3.25 = 3.614208074

Simplifying
w + 3.25 = 3.614208074

Reorder the terms:
3.25 + w = 3.614208074

Solving
3.25 + w = 3.614208074

Solving for variable 'w'.

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

Add '-3.25' to each side of the equation.
3.25 + -3.25 + w = 3.614208074 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + w = 3.614208074 + -3.25
w = 3.614208074 + -3.25

Combine like terms: 3.614208074 + -3.25 = 0.364208074
w = 0.364208074

Simplifying
w = 0.364208074

Subproblem 2
w + 3.25 = -3.614208074

Simplifying
w + 3.25 = -3.614208074

Reorder the terms:
3.25 + w = -3.614208074

Solving
3.25 + w = -3.614208074

Solving for variable 'w'.

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

Add '-3.25' to each side of the equation.
3.25 + -3.25 + w = -3.614208074 + -3.25

Combine like terms: 3.25 + -3.25 = 0.00
0.00 + w = -3.614208074 + -3.25
w = -3.614208074 + -3.25

Combine like terms: -3.614208074 + -3.25 = -6.864208074
w = -6.864208074

Simplifying
w = -6.864208074

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {0.364208074, -6.864208074}
Solution
w = {0.364208074, -6.864208074}