-4n(5n+1)=4

Solution for -4n(5n+1)=4 equation:

Simplifying
-4n(5n + 1) = 4

Reorder the terms:
-4n(1 + 5n) = 4
(1 * -4n + 5n * -4n) = 4
(-4n + -20n2) = 4

Solving
-4n + -20n2 = 4

Solving for variable 'n'.

Reorder the terms:
-4 + -4n + -20n2 = 4 + -4

Combine like terms: 4 + -4 = 0
-4 + -4n + -20n2 = 0

Factor out the Greatest Common Factor (GCF), '-4'.
-4(1 + n + 5n2) = 0

Ignore the factor -4.
Subproblem 1
Set the factor '(1 + n + 5n2)' equal to zero and attempt to solve:

Simplifying
1 + n + 5n2 = 0

Solving
1 + n + 5n2 = 0

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

Divide each side by '5'.
0.2 + 0.2n + n2 = 0

Move the constant term to the right:

Add '-0.2' to each side of the equation.
0.2 + 0.2n + -0.2 + n2 = 0 + -0.2

Reorder the terms:
0.2 + -0.2 + 0.2n + n2 = 0 + -0.2

Combine like terms: 0.2 + -0.2 = 0.0
0.0 + 0.2n + n2 = 0 + -0.2
0.2n + n2 = 0 + -0.2

Combine like terms: 0 + -0.2 = -0.2
0.2n + n2 = -0.2

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
0.2n + 0.25 + n2 = -0.2 + 0.25

Reorder the terms:
0.25 + 0.2n + n2 = -0.2 + 0.25

Combine like terms: -0.2 + 0.25 = 0.05
0.25 + 0.2n + n2 = 0.05

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 0.05

Calculate the square root of the right side: 0.223606798

Break this problem into two subproblems by setting 
(n + 0.5) equal to 0.223606798 and -0.223606798.

Subproblem 1
n + 0.5 = 0.223606798

Simplifying
n + 0.5 = 0.223606798

Reorder the terms:
0.5 + n = 0.223606798

Solving
0.5 + n = 0.223606798

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = 0.223606798 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = 0.223606798 + -0.5
n = 0.223606798 + -0.5

Combine like terms: 0.223606798 + -0.5 = -0.276393202
n = -0.276393202

Simplifying
n = -0.276393202

Subproblem 2
n + 0.5 = -0.223606798

Simplifying
n + 0.5 = -0.223606798

Reorder the terms:
0.5 + n = -0.223606798

Solving
0.5 + n = -0.223606798

Solving for variable 'n'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + n = -0.223606798 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + n = -0.223606798 + -0.5
n = -0.223606798 + -0.5

Combine like terms: -0.223606798 + -0.5 = -0.723606798
n = -0.723606798

Simplifying
n = -0.723606798

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.276393202, -0.723606798}
Solution
n = {-0.276393202, -0.723606798}