n+1=(n-2)(n-3)

Solution for n+1=(n-2)(n-3) equation:

Simplifying
n + 1 = (n + -2)(n + -3)

Reorder the terms:
1 + n = (n + -2)(n + -3)

Reorder the terms:
1 + n = (-2 + n)(n + -3)

Reorder the terms:
1 + n = (-2 + n)(-3 + n)

Multiply (-2 + n) * (-3 + n)
1 + n = (-2(-3 + n) + n(-3 + n))
1 + n = ((-3 * -2 + n * -2) + n(-3 + n))
1 + n = ((6 + -2n) + n(-3 + n))
1 + n = (6 + -2n + (-3 * n + n * n))
1 + n = (6 + -2n + (-3n + n2))

Combine like terms: -2n + -3n = -5n
1 + n = (6 + -5n + n2)

Solving
1 + n = 6 + -5n + n2

Solving for variable 'n'.

Reorder the terms:
1 + -6 + n + 5n + -1n2 = 6 + -5n + n2 + -6 + 5n + -1n2

Combine like terms: 1 + -6 = -5
-5 + n + 5n + -1n2 = 6 + -5n + n2 + -6 + 5n + -1n2

Combine like terms: n + 5n = 6n
-5 + 6n + -1n2 = 6 + -5n + n2 + -6 + 5n + -1n2

Reorder the terms:
-5 + 6n + -1n2 = 6 + -6 + -5n + 5n + n2 + -1n2

Combine like terms: 6 + -6 = 0
-5 + 6n + -1n2 = 0 + -5n + 5n + n2 + -1n2
-5 + 6n + -1n2 = -5n + 5n + n2 + -1n2

Combine like terms: -5n + 5n = 0
-5 + 6n + -1n2 = 0 + n2 + -1n2
-5 + 6n + -1n2 = n2 + -1n2

Combine like terms: n2 + -1n2 = 0
-5 + 6n + -1n2 = 0

Factor a trinomial.
(-5 + n)(1 + -1n) = 0

Subproblem 1
Set the factor '(-5 + n)' equal to zero and attempt to solve:

Simplifying
-5 + n = 0

Solving
-5 + n = 0

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

Add '5' to each side of the equation.
-5 + 5 + n = 0 + 5

Combine like terms: -5 + 5 = 0
0 + n = 0 + 5
n = 0 + 5

Combine like terms: 0 + 5 = 5
n = 5

Simplifying
n = 5
Subproblem 2
Set the factor '(1 + -1n)' equal to zero and attempt to solve:

Simplifying
1 + -1n = 0

Solving
1 + -1n = 0

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

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

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

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

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

Simplifying
n = 1
Solution
n = {5, 1}