(7n+1)=(-2n^2+n+1)

Solution for (7n+1)=(-2n^2+n+1) equation:

Simplifying
(7n + 1) = (-2n2 + n + 1)

Reorder the terms:
(1 + 7n) = (-2n2 + n + 1)

Remove parenthesis around (1 + 7n)
1 + 7n = (-2n2 + n + 1)

Reorder the terms:
1 + 7n = (1 + n + -2n2)

Remove parenthesis around (1 + n + -2n2)
1 + 7n = 1 + n + -2n2

Add '-1' to each side of the equation.
1 + -1 + 7n = 1 + n + -1 + -2n2

Combine like terms: 1 + -1 = 0
0 + 7n = 1 + n + -1 + -2n2
7n = 1 + n + -1 + -2n2

Reorder the terms:
7n = 1 + -1 + n + -2n2

Combine like terms: 1 + -1 = 0
7n = 0 + n + -2n2
7n = n + -2n2

Solving
7n = n + -2n2

Solving for variable 'n'.

Combine like terms: 7n + -1n = 6n
6n + 2n2 = n + -2n2 + -1n + 2n2

Reorder the terms:
6n + 2n2 = n + -1n + -2n2 + 2n2

Combine like terms: n + -1n = 0
6n + 2n2 = 0 + -2n2 + 2n2
6n + 2n2 = -2n2 + 2n2

Combine like terms: -2n2 + 2n2 = 0
6n + 2n2 = 0

Factor out the Greatest Common Factor (GCF), '2n'.
2n(3 + n) = 0

Ignore the factor 2.
Subproblem 1
Set the factor 'n' equal to zero and attempt to solve:

Simplifying
n = 0

Solving
n = 0

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

Simplifying
n = 0
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 = {0, -3}