(5+n)(3+n)=48

Solution for (5+n)(3+n)=48 equation:

Simplifying
(5 + n)(3 + n) = 48

Multiply (5 + n) * (3 + n)
(5(3 + n) + n(3 + n)) = 48
((3 * 5 + n * 5) + n(3 + n)) = 48
((15 + 5n) + n(3 + n)) = 48
(15 + 5n + (3 * n + n * n)) = 48
(15 + 5n + (3n + n2)) = 48

Combine like terms: 5n + 3n = 8n
(15 + 8n + n2) = 48

Solving
15 + 8n + n2 = 48

Solving for variable 'n'.

Reorder the terms:
15 + -48 + 8n + n2 = 48 + -48

Combine like terms: 15 + -48 = -33
-33 + 8n + n2 = 48 + -48

Combine like terms: 48 + -48 = 0
-33 + 8n + n2 = 0

Factor a trinomial.
(-11 + -1n)(3 + -1n) = 0

Subproblem 1
Set the factor '(-11 + -1n)' equal to zero and attempt to solve:

Simplifying
-11 + -1n = 0

Solving
-11 + -1n = 0

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

Add '11' to each side of the equation.
-11 + 11 + -1n = 0 + 11

Combine like terms: -11 + 11 = 0
0 + -1n = 0 + 11
-1n = 0 + 11

Combine like terms: 0 + 11 = 11
-1n = 11

Divide each side by '-1'.
n = -11

Simplifying
n = -11
Subproblem 2
Set the factor '(3 + -1n)' equal to zero and attempt to solve:

Simplifying
3 + -1n = 0

Solving
3 + -1n = 0

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

Add '-3' to each side of the equation.
3 + -3 + -1n = 0 + -3

Combine like terms: 3 + -3 = 0
0 + -1n = 0 + -3
-1n = 0 + -3

Combine like terms: 0 + -3 = -3
-1n = -3

Divide each side by '-1'.
n = 3

Simplifying
n = 3
Solution
n = {-11, 3}