2(w+5)5(w+2)+1=0

Solution for 2(w+5)5(w+2)+1=0 equation:

Simplifying
2(w + 5) * 5(w + 2) + 1 = 0

Reorder the terms:
2(5 + w) * 5(w + 2) + 1 = 0

Reorder the terms:
2(5 + w) * 5(2 + w) + 1 = 0

Reorder the terms for easier multiplication:
2 * 5(5 + w)(2 + w) + 1 = 0

Multiply 2 * 5
10(5 + w)(2 + w) + 1 = 0

Multiply (5 + w) * (2 + w)
10(5(2 + w) + w(2 + w)) + 1 = 0
10((2 * 5 + w * 5) + w(2 + w)) + 1 = 0
10((10 + 5w) + w(2 + w)) + 1 = 0
10(10 + 5w + (2 * w + w * w)) + 1 = 0
10(10 + 5w + (2w + w2)) + 1 = 0

Combine like terms: 5w + 2w = 7w
10(10 + 7w + w2) + 1 = 0
(10 * 10 + 7w * 10 + w2 * 10) + 1 = 0
(100 + 70w + 10w2) + 1 = 0

Reorder the terms:
100 + 1 + 70w + 10w2 = 0

Combine like terms: 100 + 1 = 101
101 + 70w + 10w2 = 0

Solving
101 + 70w + 10w2 = 0

Solving for variable 'w'.

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

Divide each side by '10'.
10.1 + 7w + w2 = 0.0

Move the constant term to the right:

Add '-10.1' to each side of the equation.
10.1 + 7w + -10.1 + w2 = 0.0 + -10.1

Reorder the terms:
10.1 + -10.1 + 7w + w2 = 0.0 + -10.1

Combine like terms: 10.1 + -10.1 = 0.0
0.0 + 7w + w2 = 0.0 + -10.1
7w + w2 = 0.0 + -10.1

Combine like terms: 0.0 + -10.1 = -10.1
7w + w2 = -10.1

The w term is 7w.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7w + 12.25 + w2 = -10.1 + 12.25

Reorder the terms:
12.25 + 7w + w2 = -10.1 + 12.25

Combine like terms: -10.1 + 12.25 = 2.15
12.25 + 7w + w2 = 2.15

Factor a perfect square on the left side:
(w + 3.5)(w + 3.5) = 2.15

Calculate the square root of the right side: 1.46628783

Break this problem into two subproblems by setting 
(w + 3.5) equal to 1.46628783 and -1.46628783.

Subproblem 1
w + 3.5 = 1.46628783

Simplifying
w + 3.5 = 1.46628783

Reorder the terms:
3.5 + w = 1.46628783

Solving
3.5 + w = 1.46628783

Solving for variable 'w'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + w = 1.46628783 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + w = 1.46628783 + -3.5
w = 1.46628783 + -3.5

Combine like terms: 1.46628783 + -3.5 = -2.03371217
w = -2.03371217

Simplifying
w = -2.03371217

Subproblem 2
w + 3.5 = -1.46628783

Simplifying
w + 3.5 = -1.46628783

Reorder the terms:
3.5 + w = -1.46628783

Solving
3.5 + w = -1.46628783

Solving for variable 'w'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + w = -1.46628783 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + w = -1.46628783 + -3.5
w = -1.46628783 + -3.5

Combine like terms: -1.46628783 + -3.5 = -4.96628783
w = -4.96628783

Simplifying
w = -4.96628783

Solution
The solution to the problem is based on the solutions
from the subproblems.
w = {-2.03371217, -4.96628783}