4(n^2)+8(n)+40=280

Solution for 4(n^2)+8(n)+40=280 equation:

Simplifying
4(n2) + 8(n) + 40 = 280

Reorder the terms:
40 + 8n + 4n2 = 280

Solving
40 + 8n + 4n2 = 280

Solving for variable 'n'.

Reorder the terms:
40 + -280 + 8n + 4n2 = 280 + -280

Combine like terms: 40 + -280 = -240
-240 + 8n + 4n2 = 280 + -280

Combine like terms: 280 + -280 = 0
-240 + 8n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-60 + 2n + n2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-60 + 2n + n2)' equal to zero and attempt to solve:

Simplifying
-60 + 2n + n2 = 0

Solving
-60 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '60' to each side of the equation.
-60 + 2n + 60 + n2 = 0 + 60

Reorder the terms:
-60 + 60 + 2n + n2 = 0 + 60

Combine like terms: -60 + 60 = 0
0 + 2n + n2 = 0 + 60
2n + n2 = 0 + 60

Combine like terms: 0 + 60 = 60
2n + n2 = 60

The n term is 2n.  Take half its coefficient (1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
2n + 1 + n2 = 60 + 1

Reorder the terms:
1 + 2n + n2 = 60 + 1

Combine like terms: 60 + 1 = 61
1 + 2n + n2 = 61

Factor a perfect square on the left side:
(n + 1)(n + 1) = 61

Calculate the square root of the right side: 7.810249676

Break this problem into two subproblems by setting 
(n + 1) equal to 7.810249676 and -7.810249676.

Subproblem 1
n + 1 = 7.810249676

Simplifying
n + 1 = 7.810249676

Reorder the terms:
1 + n = 7.810249676

Solving
1 + n = 7.810249676

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = 7.810249676 + -1

Combine like terms: 1 + -1 = 0
0 + n = 7.810249676 + -1
n = 7.810249676 + -1

Combine like terms: 7.810249676 + -1 = 6.810249676
n = 6.810249676

Simplifying
n = 6.810249676

Subproblem 2
n + 1 = -7.810249676

Simplifying
n + 1 = -7.810249676

Reorder the terms:
1 + n = -7.810249676

Solving
1 + n = -7.810249676

Solving for variable 'n'.

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

Add '-1' to each side of the equation.
1 + -1 + n = -7.810249676 + -1

Combine like terms: 1 + -1 = 0
0 + n = -7.810249676 + -1
n = -7.810249676 + -1

Combine like terms: -7.810249676 + -1 = -8.810249676
n = -8.810249676

Simplifying
n = -8.810249676

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.810249676, -8.810249676}
Solution
n = {6.810249676, -8.810249676}