2(n-1)(2n-1)+3n-1(n+1)=21

Solution for 2(n-1)(2n-1)+3n-1(n+1)=21 equation:

Simplifying
2(n + -1)(2n + -1) + 3n + -1(n + 1) = 21

Reorder the terms:
2(-1 + n)(2n + -1) + 3n + -1(n + 1) = 21

Reorder the terms:
2(-1 + n)(-1 + 2n) + 3n + -1(n + 1) = 21

Multiply (-1 + n) * (-1 + 2n)
2(-1(-1 + 2n) + n(-1 + 2n)) + 3n + -1(n + 1) = 21
2((-1 * -1 + 2n * -1) + n(-1 + 2n)) + 3n + -1(n + 1) = 21
2((1 + -2n) + n(-1 + 2n)) + 3n + -1(n + 1) = 21
2(1 + -2n + (-1 * n + 2n * n)) + 3n + -1(n + 1) = 21
2(1 + -2n + (-1n + 2n2)) + 3n + -1(n + 1) = 21

Combine like terms: -2n + -1n = -3n
2(1 + -3n + 2n2) + 3n + -1(n + 1) = 21
(1 * 2 + -3n * 2 + 2n2 * 2) + 3n + -1(n + 1) = 21
(2 + -6n + 4n2) + 3n + -1(n + 1) = 21

Reorder the terms:
2 + -6n + 4n2 + 3n + -1(1 + n) = 21
2 + -6n + 4n2 + 3n + (1 * -1 + n * -1) = 21
2 + -6n + 4n2 + 3n + (-1 + -1n) = 21

Reorder the terms:
2 + -1 + -6n + 3n + -1n + 4n2 = 21

Combine like terms: 2 + -1 = 1
1 + -6n + 3n + -1n + 4n2 = 21

Combine like terms: -6n + 3n = -3n
1 + -3n + -1n + 4n2 = 21

Combine like terms: -3n + -1n = -4n
1 + -4n + 4n2 = 21

Solving
1 + -4n + 4n2 = 21

Solving for variable 'n'.

Reorder the terms:
1 + -21 + -4n + 4n2 = 21 + -21

Combine like terms: 1 + -21 = -20
-20 + -4n + 4n2 = 21 + -21

Combine like terms: 21 + -21 = 0
-20 + -4n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-5 + -1n + n2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(-5 + -1n + n2)' equal to zero and attempt to solve:

Simplifying
-5 + -1n + n2 = 0

Solving
-5 + -1n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '5' to each side of the equation.
-5 + -1n + 5 + n2 = 0 + 5

Reorder the terms:
-5 + 5 + -1n + n2 = 0 + 5

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

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

The n term is -1n.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1n + 0.25 + n2 = 5 + 0.25

Reorder the terms:
0.25 + -1n + n2 = 5 + 0.25

Combine like terms: 5 + 0.25 = 5.25
0.25 + -1n + n2 = 5.25

Factor a perfect square on the left side:
(n + -0.5)(n + -0.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(n + -0.5) equal to 2.291287847 and -2.291287847.

Subproblem 1
n + -0.5 = 2.291287847

Simplifying
n + -0.5 = 2.291287847

Reorder the terms:
-0.5 + n = 2.291287847

Solving
-0.5 + n = 2.291287847

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = 2.291287847 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = 2.291287847 + 0.5
n = 2.291287847 + 0.5

Combine like terms: 2.291287847 + 0.5 = 2.791287847
n = 2.791287847

Simplifying
n = 2.791287847

Subproblem 2
n + -0.5 = -2.291287847

Simplifying
n + -0.5 = -2.291287847

Reorder the terms:
-0.5 + n = -2.291287847

Solving
-0.5 + n = -2.291287847

Solving for variable 'n'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + n = -2.291287847 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + n = -2.291287847 + 0.5
n = -2.291287847 + 0.5

Combine like terms: -2.291287847 + 0.5 = -1.791287847
n = -1.791287847

Simplifying
n = -1.791287847

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {2.791287847, -1.791287847}
Solution
n = {2.791287847, -1.791287847}