5c^2(c+10)-4c(2c^2-6c+1)=0

Solution for 5c^2(c+10)-4c(2c^2-6c+1)=0 equation:

Simplifying
5c2(c + 10) + -4c(2c2 + -6c + 1) = 0

Reorder the terms:
5c2(10 + c) + -4c(2c2 + -6c + 1) = 0
(10 * 5c2 + c * 5c2) + -4c(2c2 + -6c + 1) = 0
(50c2 + 5c3) + -4c(2c2 + -6c + 1) = 0

Reorder the terms:
50c2 + 5c3 + -4c(1 + -6c + 2c2) = 0
50c2 + 5c3 + (1 * -4c + -6c * -4c + 2c2 * -4c) = 0
50c2 + 5c3 + (-4c + 24c2 + -8c3) = 0

Reorder the terms:
-4c + 50c2 + 24c2 + 5c3 + -8c3 = 0

Combine like terms: 50c2 + 24c2 = 74c2
-4c + 74c2 + 5c3 + -8c3 = 0

Combine like terms: 5c3 + -8c3 = -3c3
-4c + 74c2 + -3c3 = 0

Solving
-4c + 74c2 + -3c3 = 0

Solving for variable 'c'.

Factor out the Greatest Common Factor (GCF), 'c'.
c(-4 + 74c + -3c2) = 0

Subproblem 1
Set the factor 'c' equal to zero and attempt to solve:

Simplifying
c = 0

Solving
c = 0

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

Simplifying
c = 0
Subproblem 2
Set the factor '(-4 + 74c + -3c2)' equal to zero and attempt to solve:

Simplifying
-4 + 74c + -3c2 = 0

Solving
-4 + 74c + -3c2 = 0

Begin completing the square.  Divide all terms by
-3 the coefficient of the squared term: 

Divide each side by '-3'.
1.333333333 + -24.66666667c + c2 = 0

Move the constant term to the right:

Add '-1.333333333' to each side of the equation.
1.333333333 + -24.66666667c + -1.333333333 + c2 = 0 + -1.333333333

Reorder the terms:
1.333333333 + -1.333333333 + -24.66666667c + c2 = 0 + -1.333333333

Combine like terms: 1.333333333 + -1.333333333 = 0.000000000
0.000000000 + -24.66666667c + c2 = 0 + -1.333333333
-24.66666667c + c2 = 0 + -1.333333333

Combine like terms: 0 + -1.333333333 = -1.333333333
-24.66666667c + c2 = -1.333333333

The c term is -24.66666667c.  Take half its coefficient (-12.33333334).
Square it (152.1111113) and add it to both sides.

Add '152.1111113' to each side of the equation.
-24.66666667c + 152.1111113 + c2 = -1.333333333 + 152.1111113

Reorder the terms:
152.1111113 + -24.66666667c + c2 = -1.333333333 + 152.1111113

Combine like terms: -1.333333333 + 152.1111113 = 150.777777967
152.1111113 + -24.66666667c + c2 = 150.777777967

Factor a perfect square on the left side:
(c + -12.33333334)(c + -12.33333334) = 150.777777967

Calculate the square root of the right side: 12.279160312

Break this problem into two subproblems by setting 
(c + -12.33333334) equal to 12.279160312 and -12.279160312.

Subproblem 1
c + -12.33333334 = 12.279160312

Simplifying
c + -12.33333334 = 12.279160312

Reorder the terms:
-12.33333334 + c = 12.279160312

Solving
-12.33333334 + c = 12.279160312

Solving for variable 'c'.

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

Add '12.33333334' to each side of the equation.
-12.33333334 + 12.33333334 + c = 12.279160312 + 12.33333334

Combine like terms: -12.33333334 + 12.33333334 = 0.00000000
0.00000000 + c = 12.279160312 + 12.33333334
c = 12.279160312 + 12.33333334

Combine like terms: 12.279160312 + 12.33333334 = 24.612493652
c = 24.612493652

Simplifying
c = 24.612493652

Subproblem 2
c + -12.33333334 = -12.279160312

Simplifying
c + -12.33333334 = -12.279160312

Reorder the terms:
-12.33333334 + c = -12.279160312

Solving
-12.33333334 + c = -12.279160312

Solving for variable 'c'.

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

Add '12.33333334' to each side of the equation.
-12.33333334 + 12.33333334 + c = -12.279160312 + 12.33333334

Combine like terms: -12.33333334 + 12.33333334 = 0.00000000
0.00000000 + c = -12.279160312 + 12.33333334
c = -12.279160312 + 12.33333334

Combine like terms: -12.279160312 + 12.33333334 = 0.054173028
c = 0.054173028

Simplifying
c = 0.054173028

Solution
The solution to the problem is based on the solutions
from the subproblems.
c = {24.612493652, 0.054173028}
Solution
c = {0, 24.612493652, 0.054173028}