2(n+2)(n+1)=24n(4n-1)

Solution for 2(n+2)(n+1)=24n(4n-1) equation:

Simplifying
2(n + 2)(n + 1) = 24n(4n + -1)

Reorder the terms:
2(2 + n)(n + 1) = 24n(4n + -1)

Reorder the terms:
2(2 + n)(1 + n) = 24n(4n + -1)

Multiply (2 + n) * (1 + n)
2(2(1 + n) + n(1 + n)) = 24n(4n + -1)
2((1 * 2 + n * 2) + n(1 + n)) = 24n(4n + -1)
2((2 + 2n) + n(1 + n)) = 24n(4n + -1)
2(2 + 2n + (1 * n + n * n)) = 24n(4n + -1)
2(2 + 2n + (1n + n2)) = 24n(4n + -1)

Combine like terms: 2n + 1n = 3n
2(2 + 3n + n2) = 24n(4n + -1)
(2 * 2 + 3n * 2 + n2 * 2) = 24n(4n + -1)
(4 + 6n + 2n2) = 24n(4n + -1)

Reorder the terms:
4 + 6n + 2n2 = 24n(-1 + 4n)
4 + 6n + 2n2 = (-1 * 24n + 4n * 24n)
4 + 6n + 2n2 = (-24n + 96n2)

Solving
4 + 6n + 2n2 = -24n + 96n2

Solving for variable 'n'.

Reorder the terms:
4 + 6n + 24n + 2n2 + -96n2 = -24n + 96n2 + 24n + -96n2

Combine like terms: 6n + 24n = 30n
4 + 30n + 2n2 + -96n2 = -24n + 96n2 + 24n + -96n2

Combine like terms: 2n2 + -96n2 = -94n2
4 + 30n + -94n2 = -24n + 96n2 + 24n + -96n2

Reorder the terms:
4 + 30n + -94n2 = -24n + 24n + 96n2 + -96n2

Combine like terms: -24n + 24n = 0
4 + 30n + -94n2 = 0 + 96n2 + -96n2
4 + 30n + -94n2 = 96n2 + -96n2

Combine like terms: 96n2 + -96n2 = 0
4 + 30n + -94n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(2 + 15n + -47n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(2 + 15n + -47n2)' equal to zero and attempt to solve:

Simplifying
2 + 15n + -47n2 = 0

Solving
2 + 15n + -47n2 = 0

Begin completing the square.  Divide all terms by
-47 the coefficient of the squared term: 

Divide each side by '-47'.
-0.04255319149 + -0.3191489362n + n2 = 0

Move the constant term to the right:

Add '0.04255319149' to each side of the equation.
-0.04255319149 + -0.3191489362n + 0.04255319149 + n2 = 0 + 0.04255319149

Reorder the terms:
-0.04255319149 + 0.04255319149 + -0.3191489362n + n2 = 0 + 0.04255319149

Combine like terms: -0.04255319149 + 0.04255319149 = 0.00000000000
0.00000000000 + -0.3191489362n + n2 = 0 + 0.04255319149
-0.3191489362n + n2 = 0 + 0.04255319149

Combine like terms: 0 + 0.04255319149 = 0.04255319149
-0.3191489362n + n2 = 0.04255319149

The n term is -0.3191489362n.  Take half its coefficient (-0.1595744681).
Square it (0.02546401087) and add it to both sides.

Add '0.02546401087' to each side of the equation.
-0.3191489362n + 0.02546401087 + n2 = 0.04255319149 + 0.02546401087

Reorder the terms:
0.02546401087 + -0.3191489362n + n2 = 0.04255319149 + 0.02546401087

Combine like terms: 0.04255319149 + 0.02546401087 = 0.06801720236
0.02546401087 + -0.3191489362n + n2 = 0.06801720236

Factor a perfect square on the left side:
(n + -0.1595744681)(n + -0.1595744681) = 0.06801720236

Calculate the square root of the right side: 0.260801078

Break this problem into two subproblems by setting 
(n + -0.1595744681) equal to 0.260801078 and -0.260801078.

Subproblem 1
n + -0.1595744681 = 0.260801078

Simplifying
n + -0.1595744681 = 0.260801078

Reorder the terms:
-0.1595744681 + n = 0.260801078

Solving
-0.1595744681 + n = 0.260801078

Solving for variable 'n'.

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

Add '0.1595744681' to each side of the equation.
-0.1595744681 + 0.1595744681 + n = 0.260801078 + 0.1595744681

Combine like terms: -0.1595744681 + 0.1595744681 = 0.0000000000
0.0000000000 + n = 0.260801078 + 0.1595744681
n = 0.260801078 + 0.1595744681

Combine like terms: 0.260801078 + 0.1595744681 = 0.4203755461
n = 0.4203755461

Simplifying
n = 0.4203755461

Subproblem 2
n + -0.1595744681 = -0.260801078

Simplifying
n + -0.1595744681 = -0.260801078

Reorder the terms:
-0.1595744681 + n = -0.260801078

Solving
-0.1595744681 + n = -0.260801078

Solving for variable 'n'.

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

Add '0.1595744681' to each side of the equation.
-0.1595744681 + 0.1595744681 + n = -0.260801078 + 0.1595744681

Combine like terms: -0.1595744681 + 0.1595744681 = 0.0000000000
0.0000000000 + n = -0.260801078 + 0.1595744681
n = -0.260801078 + 0.1595744681

Combine like terms: -0.260801078 + 0.1595744681 = -0.1012266099
n = -0.1012266099

Simplifying
n = -0.1012266099

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {0.4203755461, -0.1012266099}
Solution
n = {0.4203755461, -0.1012266099}