(w+1)(2w+3)=102

Solution for (w+1)(2w+3)=102 equation:

Simplifying
(w + 1)(2w + 3) = 102

Reorder the terms:
(1 + w)(2w + 3) = 102

Reorder the terms:
(1 + w)(3 + 2w) = 102

Multiply (1 + w) * (3 + 2w)
(1(3 + 2w) + w(3 + 2w)) = 102
((3 * 1 + 2w * 1) + w(3 + 2w)) = 102
((3 + 2w) + w(3 + 2w)) = 102
(3 + 2w + (3 * w + 2w * w)) = 102
(3 + 2w + (3w + 2w2)) = 102

Combine like terms: 2w + 3w = 5w
(3 + 5w + 2w2) = 102

Solving
3 + 5w + 2w2 = 102

Solving for variable 'w'.

Reorder the terms:
3 + -102 + 5w + 2w2 = 102 + -102

Combine like terms: 3 + -102 = -99
-99 + 5w + 2w2 = 102 + -102

Combine like terms: 102 + -102 = 0
-99 + 5w + 2w2 = 0

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

Divide each side by '2'.
-49.5 + 2.5w + w2 = 0

Move the constant term to the right:

Add '49.5' to each side of the equation.
-49.5 + 2.5w + 49.5 + w2 = 0 + 49.5

Reorder the terms:
-49.5 + 49.5 + 2.5w + w2 = 0 + 49.5

Combine like terms: -49.5 + 49.5 = 0.0
0.0 + 2.5w + w2 = 0 + 49.5
2.5w + w2 = 0 + 49.5

Combine like terms: 0 + 49.5 = 49.5
2.5w + w2 = 49.5

The w term is 2.5w.  Take half its coefficient (1.25).
Square it (1.5625) and add it to both sides.

Add '1.5625' to each side of the equation.
2.5w + 1.5625 + w2 = 49.5 + 1.5625

Reorder the terms:
1.5625 + 2.5w + w2 = 49.5 + 1.5625

Combine like terms: 49.5 + 1.5625 = 51.0625
1.5625 + 2.5w + w2 = 51.0625

Factor a perfect square on the left side:
(w + 1.25)(w + 1.25) = 51.0625

Calculate the square root of the right side: 7.145802964

Break this problem into two subproblems by setting 
(w + 1.25) equal to 7.145802964 and -7.145802964.

Subproblem 1
w + 1.25 = 7.145802964

Simplifying
w + 1.25 = 7.145802964

Reorder the terms:
1.25 + w = 7.145802964

Solving
1.25 + w = 7.145802964

Solving for variable 'w'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + w = 7.145802964 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + w = 7.145802964 + -1.25
w = 7.145802964 + -1.25

Combine like terms: 7.145802964 + -1.25 = 5.895802964
w = 5.895802964

Simplifying
w = 5.895802964

Subproblem 2
w + 1.25 = -7.145802964

Simplifying
w + 1.25 = -7.145802964

Reorder the terms:
1.25 + w = -7.145802964

Solving
1.25 + w = -7.145802964

Solving for variable 'w'.

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

Add '-1.25' to each side of the equation.
1.25 + -1.25 + w = -7.145802964 + -1.25

Combine like terms: 1.25 + -1.25 = 0.00
0.00 + w = -7.145802964 + -1.25
w = -7.145802964 + -1.25

Combine like terms: -7.145802964 + -1.25 = -8.395802964
w = -8.395802964

Simplifying
w = -8.395802964

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {5.895802964, -8.395802964}