(n+3)(n-5)=90

Solution for (n+3)(n-5)=90 equation:

Simplifying
(n + 3)(n + -5) = 90

Reorder the terms:
(3 + n)(n + -5) = 90

Reorder the terms:
(3 + n)(-5 + n) = 90

Multiply (3 + n) * (-5 + n)
(3(-5 + n) + n(-5 + n)) = 90
((-5 * 3 + n * 3) + n(-5 + n)) = 90
((-15 + 3n) + n(-5 + n)) = 90
(-15 + 3n + (-5 * n + n * n)) = 90
(-15 + 3n + (-5n + n2)) = 90

Combine like terms: 3n + -5n = -2n
(-15 + -2n + n2) = 90

Solving
-15 + -2n + n2 = 90

Solving for variable 'n'.

Reorder the terms:
-15 + -90 + -2n + n2 = 90 + -90

Combine like terms: -15 + -90 = -105
-105 + -2n + n2 = 90 + -90

Combine like terms: 90 + -90 = 0
-105 + -2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '105' to each side of the equation.
-105 + -2n + 105 + n2 = 0 + 105

Reorder the terms:
-105 + 105 + -2n + n2 = 0 + 105

Combine like terms: -105 + 105 = 0
0 + -2n + n2 = 0 + 105
-2n + n2 = 0 + 105

Combine like terms: 0 + 105 = 105
-2n + n2 = 105

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 = 105 + 1

Reorder the terms:
1 + -2n + n2 = 105 + 1

Combine like terms: 105 + 1 = 106
1 + -2n + n2 = 106

Factor a perfect square on the left side:
(n + -1)(n + -1) = 106

Calculate the square root of the right side: 10.295630141

Break this problem into two subproblems by setting 
(n + -1) equal to 10.295630141 and -10.295630141.

Subproblem 1
n + -1 = 10.295630141

Simplifying
n + -1 = 10.295630141

Reorder the terms:
-1 + n = 10.295630141

Solving
-1 + n = 10.295630141

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 = 10.295630141 + 1

Combine like terms: -1 + 1 = 0
0 + n = 10.295630141 + 1
n = 10.295630141 + 1

Combine like terms: 10.295630141 + 1 = 11.295630141
n = 11.295630141

Simplifying
n = 11.295630141

Subproblem 2
n + -1 = -10.295630141

Simplifying
n + -1 = -10.295630141

Reorder the terms:
-1 + n = -10.295630141

Solving
-1 + n = -10.295630141

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 = -10.295630141 + 1

Combine like terms: -1 + 1 = 0
0 + n = -10.295630141 + 1
n = -10.295630141 + 1

Combine like terms: -10.295630141 + 1 = -9.295630141
n = -9.295630141

Simplifying
n = -9.295630141

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {11.295630141, -9.295630141}