5s(100+2s)=4s-2(10+5)

Solution for 5s(100+2s)=4s-2(10+5) equation:

Simplifying
5s(100 + 2s) = 4s + -2(10 + 5)
(100 * 5s + 2s * 5s) = 4s + -2(10 + 5)
(500s + 10s2) = 4s + -2(10 + 5)

Combine like terms: 10 + 5 = 15
500s + 10s2 = 4s + -2(15)

Multiply -2 * 15
500s + 10s2 = 4s + -30

Reorder the terms:
500s + 10s2 = -30 + 4s

Solving
500s + 10s2 = -30 + 4s

Solving for variable 's'.

Reorder the terms:
30 + 500s + -4s + 10s2 = -30 + 4s + 30 + -4s

Combine like terms: 500s + -4s = 496s
30 + 496s + 10s2 = -30 + 4s + 30 + -4s

Reorder the terms:
30 + 496s + 10s2 = -30 + 30 + 4s + -4s

Combine like terms: -30 + 30 = 0
30 + 496s + 10s2 = 0 + 4s + -4s
30 + 496s + 10s2 = 4s + -4s

Combine like terms: 4s + -4s = 0
30 + 496s + 10s2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(15 + 248s + 5s2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(15 + 248s + 5s2)' equal to zero and attempt to solve:

Simplifying
15 + 248s + 5s2 = 0

Solving
15 + 248s + 5s2 = 0

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

Divide each side by '5'.
3 + 49.6s + s2 = 0

Move the constant term to the right:

Add '-3' to each side of the equation.
3 + 49.6s + -3 + s2 = 0 + -3

Reorder the terms:
3 + -3 + 49.6s + s2 = 0 + -3

Combine like terms: 3 + -3 = 0
0 + 49.6s + s2 = 0 + -3
49.6s + s2 = 0 + -3

Combine like terms: 0 + -3 = -3
49.6s + s2 = -3

The s term is 49.6s.  Take half its coefficient (24.8).
Square it (615.04) and add it to both sides.

Add '615.04' to each side of the equation.
49.6s + 615.04 + s2 = -3 + 615.04

Reorder the terms:
615.04 + 49.6s + s2 = -3 + 615.04

Combine like terms: -3 + 615.04 = 612.04
615.04 + 49.6s + s2 = 612.04

Factor a perfect square on the left side:
(s + 24.8)(s + 24.8) = 612.04

Calculate the square root of the right side: 24.739442193

Break this problem into two subproblems by setting 
(s + 24.8) equal to 24.739442193 and -24.739442193.

Subproblem 1
s + 24.8 = 24.739442193

Simplifying
s + 24.8 = 24.739442193

Reorder the terms:
24.8 + s = 24.739442193

Solving
24.8 + s = 24.739442193

Solving for variable 's'.

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

Add '-24.8' to each side of the equation.
24.8 + -24.8 + s = 24.739442193 + -24.8

Combine like terms: 24.8 + -24.8 = 0.0
0.0 + s = 24.739442193 + -24.8
s = 24.739442193 + -24.8

Combine like terms: 24.739442193 + -24.8 = -0.060557807
s = -0.060557807

Simplifying
s = -0.060557807

Subproblem 2
s + 24.8 = -24.739442193

Simplifying
s + 24.8 = -24.739442193

Reorder the terms:
24.8 + s = -24.739442193

Solving
24.8 + s = -24.739442193

Solving for variable 's'.

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

Add '-24.8' to each side of the equation.
24.8 + -24.8 + s = -24.739442193 + -24.8

Combine like terms: 24.8 + -24.8 = 0.0
0.0 + s = -24.739442193 + -24.8
s = -24.739442193 + -24.8

Combine like terms: -24.739442193 + -24.8 = -49.539442193
s = -49.539442193

Simplifying
s = -49.539442193

Solution
The solution to the problem is based on the solutions
from the subproblems.
s = {-0.060557807, -49.539442193}
Solution
s = {-0.060557807, -49.539442193}