(3+w)w+w+(3+w)=24

Solution for (3+w)w+w+(3+w)=24 equation:

Simplifying
(3 + w) * w + w + (3 + w) = 24

Reorder the terms for easier multiplication:
w(3 + w) + w + (3 + w) = 24
(3 * w + w * w) + w + (3 + w) = 24
(3w + w2) + w + (3 + w) = 24

Remove parenthesis around (3 + w)
3w + w2 + w + 3 + w = 24

Reorder the terms:
3 + 3w + w + w + w2 = 24

Combine like terms: 3w + w = 4w
3 + 4w + w + w2 = 24

Combine like terms: 4w + w = 5w
3 + 5w + w2 = 24

Solving
3 + 5w + w2 = 24

Solving for variable 'w'.

Reorder the terms:
3 + -24 + 5w + w2 = 24 + -24

Combine like terms: 3 + -24 = -21
-21 + 5w + w2 = 24 + -24

Combine like terms: 24 + -24 = 0
-21 + 5w + w2 = 0

Begin completing the square.

Move the constant term to the right:

Add '21' to each side of the equation.
-21 + 5w + 21 + w2 = 0 + 21

Reorder the terms:
-21 + 21 + 5w + w2 = 0 + 21

Combine like terms: -21 + 21 = 0
0 + 5w + w2 = 0 + 21
5w + w2 = 0 + 21

Combine like terms: 0 + 21 = 21
5w + w2 = 21

The w term is 5w.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5w + 6.25 + w2 = 21 + 6.25

Reorder the terms:
6.25 + 5w + w2 = 21 + 6.25

Combine like terms: 21 + 6.25 = 27.25
6.25 + 5w + w2 = 27.25

Factor a perfect square on the left side:
(w + 2.5)(w + 2.5) = 27.25

Calculate the square root of the right side: 5.220153254

Break this problem into two subproblems by setting 
(w + 2.5) equal to 5.220153254 and -5.220153254.

Subproblem 1
w + 2.5 = 5.220153254

Simplifying
w + 2.5 = 5.220153254

Reorder the terms:
2.5 + w = 5.220153254

Solving
2.5 + w = 5.220153254

Solving for variable 'w'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + w = 5.220153254 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + w = 5.220153254 + -2.5
w = 5.220153254 + -2.5

Combine like terms: 5.220153254 + -2.5 = 2.720153254
w = 2.720153254

Simplifying
w = 2.720153254

Subproblem 2
w + 2.5 = -5.220153254

Simplifying
w + 2.5 = -5.220153254

Reorder the terms:
2.5 + w = -5.220153254

Solving
2.5 + w = -5.220153254

Solving for variable 'w'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + w = -5.220153254 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + w = -5.220153254 + -2.5
w = -5.220153254 + -2.5

Combine like terms: -5.220153254 + -2.5 = -7.720153254
w = -7.720153254

Simplifying
w = -7.720153254

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {2.720153254, -7.720153254}