-4n(n+7)+8(1-7n)=

Solution for -4n(n+7)+8(1-7n)= equation:

Simplifying
-4n(n + 7) + 8(1 + -7n) = 0

Reorder the terms:
-4n(7 + n) + 8(1 + -7n) = 0
(7 * -4n + n * -4n) + 8(1 + -7n) = 0
(-28n + -4n2) + 8(1 + -7n) = 0
-28n + -4n2 + (1 * 8 + -7n * 8) = 0
-28n + -4n2 + (8 + -56n) = 0

Reorder the terms:
8 + -28n + -56n + -4n2 = 0

Combine like terms: -28n + -56n = -84n
8 + -84n + -4n2 = 0

Solving
8 + -84n + -4n2 = 0

Solving for variable 'n'.

Factor out the Greatest Common Factor (GCF), '4'.
4(2 + -21n + -1n2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(2 + -21n + -1n2)' equal to zero and attempt to solve:

Simplifying
2 + -21n + -1n2 = 0

Solving
2 + -21n + -1n2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-2 + 21n + n2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + 21n + 2 + n2 = 0 + 2

Reorder the terms:
-2 + 2 + 21n + n2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + 21n + n2 = 0 + 2
21n + n2 = 0 + 2

Combine like terms: 0 + 2 = 2
21n + n2 = 2

The n term is 21n.  Take half its coefficient (10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
21n + 110.25 + n2 = 2 + 110.25

Reorder the terms:
110.25 + 21n + n2 = 2 + 110.25

Combine like terms: 2 + 110.25 = 112.25
110.25 + 21n + n2 = 112.25

Factor a perfect square on the left side:
(n + 10.5)(n + 10.5) = 112.25

Calculate the square root of the right side: 10.59481005

Break this problem into two subproblems by setting 
(n + 10.5) equal to 10.59481005 and -10.59481005.

Subproblem 1
n + 10.5 = 10.59481005

Simplifying
n + 10.5 = 10.59481005

Reorder the terms:
10.5 + n = 10.59481005

Solving
10.5 + n = 10.59481005

Solving for variable 'n'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + n = 10.59481005 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + n = 10.59481005 + -10.5
n = 10.59481005 + -10.5

Combine like terms: 10.59481005 + -10.5 = 0.09481005
n = 0.09481005

Simplifying
n = 0.09481005

Subproblem 2
n + 10.5 = -10.59481005

Simplifying
n + 10.5 = -10.59481005

Reorder the terms:
10.5 + n = -10.59481005

Solving
10.5 + n = -10.59481005

Solving for variable 'n'.

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

Add '-10.5' to each side of the equation.
10.5 + -10.5 + n = -10.59481005 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + n = -10.59481005 + -10.5
n = -10.59481005 + -10.5

Combine like terms: -10.59481005 + -10.5 = -21.09481005
n = -21.09481005

Simplifying
n = -21.09481005

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.09481005, -21.09481005}
Solution
n = {0.09481005, -21.09481005}