4(n+2)(n+5)=230

Solution for 4(n+2)(n+5)=230 equation:

Simplifying
4(n + 2)(n + 5) = 230

Reorder the terms:
4(2 + n)(n + 5) = 230

Reorder the terms:
4(2 + n)(5 + n) = 230

Multiply (2 + n) * (5 + n)
4(2(5 + n) + n(5 + n)) = 230
4((5 * 2 + n * 2) + n(5 + n)) = 230
4((10 + 2n) + n(5 + n)) = 230
4(10 + 2n + (5 * n + n * n)) = 230
4(10 + 2n + (5n + n2)) = 230

Combine like terms: 2n + 5n = 7n
4(10 + 7n + n2) = 230
(10 * 4 + 7n * 4 + n2 * 4) = 230
(40 + 28n + 4n2) = 230

Solving
40 + 28n + 4n2 = 230

Solving for variable 'n'.

Reorder the terms:
40 + -230 + 28n + 4n2 = 230 + -230

Combine like terms: 40 + -230 = -190
-190 + 28n + 4n2 = 230 + -230

Combine like terms: 230 + -230 = 0
-190 + 28n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-95 + 14n + 2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-95 + 14n + 2n2)' equal to zero and attempt to solve:

Simplifying
-95 + 14n + 2n2 = 0

Solving
-95 + 14n + 2n2 = 0

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

Divide each side by '2'.
-47.5 + 7n + n2 = 0

Move the constant term to the right:

Add '47.5' to each side of the equation.
-47.5 + 7n + 47.5 + n2 = 0 + 47.5

Reorder the terms:
-47.5 + 47.5 + 7n + n2 = 0 + 47.5

Combine like terms: -47.5 + 47.5 = 0.0
0.0 + 7n + n2 = 0 + 47.5
7n + n2 = 0 + 47.5

Combine like terms: 0 + 47.5 = 47.5
7n + n2 = 47.5

The n term is 7n.  Take half its coefficient (3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
7n + 12.25 + n2 = 47.5 + 12.25

Reorder the terms:
12.25 + 7n + n2 = 47.5 + 12.25

Combine like terms: 47.5 + 12.25 = 59.75
12.25 + 7n + n2 = 59.75

Factor a perfect square on the left side:
(n + 3.5)(n + 3.5) = 59.75

Calculate the square root of the right side: 7.729812417

Break this problem into two subproblems by setting 
(n + 3.5) equal to 7.729812417 and -7.729812417.

Subproblem 1
n + 3.5 = 7.729812417

Simplifying
n + 3.5 = 7.729812417

Reorder the terms:
3.5 + n = 7.729812417

Solving
3.5 + n = 7.729812417

Solving for variable 'n'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + n = 7.729812417 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + n = 7.729812417 + -3.5
n = 7.729812417 + -3.5

Combine like terms: 7.729812417 + -3.5 = 4.229812417
n = 4.229812417

Simplifying
n = 4.229812417

Subproblem 2
n + 3.5 = -7.729812417

Simplifying
n + 3.5 = -7.729812417

Reorder the terms:
3.5 + n = -7.729812417

Solving
3.5 + n = -7.729812417

Solving for variable 'n'.

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

Add '-3.5' to each side of the equation.
3.5 + -3.5 + n = -7.729812417 + -3.5

Combine like terms: 3.5 + -3.5 = 0.0
0.0 + n = -7.729812417 + -3.5
n = -7.729812417 + -3.5

Combine like terms: -7.729812417 + -3.5 = -11.229812417
n = -11.229812417

Simplifying
n = -11.229812417

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {4.229812417, -11.229812417}
Solution
n = {4.229812417, -11.229812417}