4n^2+8n+12=280

Solution for 4n^2+8n+12=280 equation:

Simplifying
4n2 + 8n + 12 = 280

Reorder the terms:
12 + 8n + 4n2 = 280

Solving
12 + 8n + 4n2 = 280

Solving for variable 'n'.

Reorder the terms:
12 + -280 + 8n + 4n2 = 280 + -280

Combine like terms: 12 + -280 = -268
-268 + 8n + 4n2 = 280 + -280

Combine like terms: 280 + -280 = 0
-268 + 8n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-67 + 2n + n2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-67 + 2n + n2)' equal to zero and attempt to solve:

Simplifying
-67 + 2n + n2 = 0

Solving
-67 + 2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '67' to each side of the equation.
-67 + 2n + 67 + n2 = 0 + 67

Reorder the terms:
-67 + 67 + 2n + n2 = 0 + 67

Combine like terms: -67 + 67 = 0
0 + 2n + n2 = 0 + 67
2n + n2 = 0 + 67

Combine like terms: 0 + 67 = 67
2n + n2 = 67

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 = 67 + 1

Reorder the terms:
1 + 2n + n2 = 67 + 1

Combine like terms: 67 + 1 = 68
1 + 2n + n2 = 68

Factor a perfect square on the left side:
(n + 1)(n + 1) = 68

Calculate the square root of the right side: 8.246211251

Break this problem into two subproblems by setting 
(n + 1) equal to 8.246211251 and -8.246211251.

Subproblem 1
n + 1 = 8.246211251

Simplifying
n + 1 = 8.246211251

Reorder the terms:
1 + n = 8.246211251

Solving
1 + n = 8.246211251

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 = 8.246211251 + -1

Combine like terms: 1 + -1 = 0
0 + n = 8.246211251 + -1
n = 8.246211251 + -1

Combine like terms: 8.246211251 + -1 = 7.246211251
n = 7.246211251

Simplifying
n = 7.246211251

Subproblem 2
n + 1 = -8.246211251

Simplifying
n + 1 = -8.246211251

Reorder the terms:
1 + n = -8.246211251

Solving
1 + n = -8.246211251

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 = -8.246211251 + -1

Combine like terms: 1 + -1 = 0
0 + n = -8.246211251 + -1
n = -8.246211251 + -1

Combine like terms: -8.246211251 + -1 = -9.246211251
n = -9.246211251

Simplifying
n = -9.246211251

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {7.246211251, -9.246211251}
Solution
n = {7.246211251, -9.246211251}