(4-n)(4+n)=7(n+4)

Solution for (4-n)(4+n)=7(n+4) equation:

Simplifying
(4 + -1n)(4 + n) = 7(n + 4)

Multiply (4 + -1n) * (4 + n)
(4(4 + n) + -1n * (4 + n)) = 7(n + 4)
((4 * 4 + n * 4) + -1n * (4 + n)) = 7(n + 4)
((16 + 4n) + -1n * (4 + n)) = 7(n + 4)
(16 + 4n + (4 * -1n + n * -1n)) = 7(n + 4)
(16 + 4n + (-4n + -1n2)) = 7(n + 4)

Combine like terms: 4n + -4n = 0
(16 + 0 + -1n2) = 7(n + 4)
(16 + -1n2) = 7(n + 4)

Reorder the terms:
16 + -1n2 = 7(4 + n)
16 + -1n2 = (4 * 7 + n * 7)
16 + -1n2 = (28 + 7n)

Solving
16 + -1n2 = 28 + 7n

Solving for variable 'n'.

Reorder the terms:
16 + -28 + -7n + -1n2 = 28 + 7n + -28 + -7n

Combine like terms: 16 + -28 = -12
-12 + -7n + -1n2 = 28 + 7n + -28 + -7n

Reorder the terms:
-12 + -7n + -1n2 = 28 + -28 + 7n + -7n

Combine like terms: 28 + -28 = 0
-12 + -7n + -1n2 = 0 + 7n + -7n
-12 + -7n + -1n2 = 7n + -7n

Combine like terms: 7n + -7n = 0
-12 + -7n + -1n2 = 0

Factor out the Greatest Common Factor (GCF), '-1'.
-1(12 + 7n + n2) = 0

Factor a trinomial.
-1((4 + n)(3 + n)) = 0

Ignore the factor -1.
Subproblem 1
Set the factor '(4 + n)' equal to zero and attempt to solve:

Simplifying
4 + n = 0

Solving
4 + n = 0

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

Add '-4' to each side of the equation.
4 + -4 + n = 0 + -4

Combine like terms: 4 + -4 = 0
0 + n = 0 + -4
n = 0 + -4

Combine like terms: 0 + -4 = -4
n = -4

Simplifying
n = -4
Subproblem 2
Set the factor '(3 + n)' equal to zero and attempt to solve:

Simplifying
3 + n = 0

Solving
3 + n = 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 + n = 0 + -3

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

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

Simplifying
n = -3
Solution
n = {-4, -3}