5=(n+3)*(n+12)

Solution for 5=(n+3)*(n+12) equation:

Simplifying
5 = (n + 3)(n + 12)

Reorder the terms:
5 = (3 + n)(n + 12)

Reorder the terms:
5 = (3 + n)(12 + n)

Multiply (3 + n) * (12 + n)
5 = (3(12 + n) + n(12 + n))
5 = ((12 * 3 + n * 3) + n(12 + n))
5 = ((36 + 3n) + n(12 + n))
5 = (36 + 3n + (12 * n + n * n))
5 = (36 + 3n + (12n + n2))

Combine like terms: 3n + 12n = 15n
5 = (36 + 15n + n2)

Solving
5 = 36 + 15n + n2

Solving for variable 'n'.

Combine like terms: 5 + -36 = -31
-31 + -15n + -1n2 = 36 + 15n + n2 + -36 + -15n + -1n2

Reorder the terms:
-31 + -15n + -1n2 = 36 + -36 + 15n + -15n + n2 + -1n2

Combine like terms: 36 + -36 = 0
-31 + -15n + -1n2 = 0 + 15n + -15n + n2 + -1n2
-31 + -15n + -1n2 = 15n + -15n + n2 + -1n2

Combine like terms: 15n + -15n = 0
-31 + -15n + -1n2 = 0 + n2 + -1n2
-31 + -15n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-31 + -15n + -1n2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(31 + 15n + n2) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(31 + 15n + n2)' equal to zero and attempt to solve:

Simplifying
31 + 15n + n2 = 0

Solving
31 + 15n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 15n + -31 + n2 = 0 + -31

Reorder the terms:
31 + -31 + 15n + n2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 15n + n2 = 0 + -31
15n + n2 = 0 + -31

Combine like terms: 0 + -31 = -31
15n + n2 = -31

The n term is 15n.  Take half its coefficient (7.5).
Square it (56.25) and add it to both sides.

Add '56.25' to each side of the equation.
15n + 56.25 + n2 = -31 + 56.25

Reorder the terms:
56.25 + 15n + n2 = -31 + 56.25

Combine like terms: -31 + 56.25 = 25.25
56.25 + 15n + n2 = 25.25

Factor a perfect square on the left side:
(n + 7.5)(n + 7.5) = 25.25

Calculate the square root of the right side: 5.024937811

Break this problem into two subproblems by setting 
(n + 7.5) equal to 5.024937811 and -5.024937811.

Subproblem 1
n + 7.5 = 5.024937811

Simplifying
n + 7.5 = 5.024937811

Reorder the terms:
7.5 + n = 5.024937811

Solving
7.5 + n = 5.024937811

Solving for variable 'n'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + n = 5.024937811 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + n = 5.024937811 + -7.5
n = 5.024937811 + -7.5

Combine like terms: 5.024937811 + -7.5 = -2.475062189
n = -2.475062189

Simplifying
n = -2.475062189

Subproblem 2
n + 7.5 = -5.024937811

Simplifying
n + 7.5 = -5.024937811

Reorder the terms:
7.5 + n = -5.024937811

Solving
7.5 + n = -5.024937811

Solving for variable 'n'.

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

Add '-7.5' to each side of the equation.
7.5 + -7.5 + n = -5.024937811 + -7.5

Combine like terms: 7.5 + -7.5 = 0.0
0.0 + n = -5.024937811 + -7.5
n = -5.024937811 + -7.5

Combine like terms: -5.024937811 + -7.5 = -12.524937811
n = -12.524937811

Simplifying
n = -12.524937811

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-2.475062189, -12.524937811}
Solution
n = {-2.475062189, -12.524937811}