(n+3)(n+5)=49+10n

Solution for (n+3)(n+5)=49+10n equation:

Simplifying
(n + 3)(n + 5) = 49 + 10n

Reorder the terms:
(3 + n)(n + 5) = 49 + 10n

Reorder the terms:
(3 + n)(5 + n) = 49 + 10n

Multiply (3 + n) * (5 + n)
(3(5 + n) + n(5 + n)) = 49 + 10n
((5 * 3 + n * 3) + n(5 + n)) = 49 + 10n
((15 + 3n) + n(5 + n)) = 49 + 10n
(15 + 3n + (5 * n + n * n)) = 49 + 10n
(15 + 3n + (5n + n2)) = 49 + 10n

Combine like terms: 3n + 5n = 8n
(15 + 8n + n2) = 49 + 10n

Solving
15 + 8n + n2 = 49 + 10n

Solving for variable 'n'.

Reorder the terms:
15 + -49 + 8n + -10n + n2 = 49 + 10n + -49 + -10n

Combine like terms: 15 + -49 = -34
-34 + 8n + -10n + n2 = 49 + 10n + -49 + -10n

Combine like terms: 8n + -10n = -2n
-34 + -2n + n2 = 49 + 10n + -49 + -10n

Reorder the terms:
-34 + -2n + n2 = 49 + -49 + 10n + -10n

Combine like terms: 49 + -49 = 0
-34 + -2n + n2 = 0 + 10n + -10n
-34 + -2n + n2 = 10n + -10n

Combine like terms: 10n + -10n = 0
-34 + -2n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '34' to each side of the equation.
-34 + -2n + 34 + n2 = 0 + 34

Reorder the terms:
-34 + 34 + -2n + n2 = 0 + 34

Combine like terms: -34 + 34 = 0
0 + -2n + n2 = 0 + 34
-2n + n2 = 0 + 34

Combine like terms: 0 + 34 = 34
-2n + n2 = 34

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 = 34 + 1

Reorder the terms:
1 + -2n + n2 = 34 + 1

Combine like terms: 34 + 1 = 35
1 + -2n + n2 = 35

Factor a perfect square on the left side:
(n + -1)(n + -1) = 35

Calculate the square root of the right side: 5.916079783

Break this problem into two subproblems by setting 
(n + -1) equal to 5.916079783 and -5.916079783.

Subproblem 1
n + -1 = 5.916079783

Simplifying
n + -1 = 5.916079783

Reorder the terms:
-1 + n = 5.916079783

Solving
-1 + n = 5.916079783

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 = 5.916079783 + 1

Combine like terms: -1 + 1 = 0
0 + n = 5.916079783 + 1
n = 5.916079783 + 1

Combine like terms: 5.916079783 + 1 = 6.916079783
n = 6.916079783

Simplifying
n = 6.916079783

Subproblem 2
n + -1 = -5.916079783

Simplifying
n + -1 = -5.916079783

Reorder the terms:
-1 + n = -5.916079783

Solving
-1 + n = -5.916079783

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 = -5.916079783 + 1

Combine like terms: -1 + 1 = 0
0 + n = -5.916079783 + 1
n = -5.916079783 + 1

Combine like terms: -5.916079783 + 1 = -4.916079783
n = -4.916079783

Simplifying
n = -4.916079783

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {6.916079783, -4.916079783}