(2n+4)(2n-1)-1=19

Solution for (2n+4)(2n-1)-1=19 equation:

Simplifying
(2n + 4)(2n + -1) + -1 = 19

Reorder the terms:
(4 + 2n)(2n + -1) + -1 = 19

Reorder the terms:
(4 + 2n)(-1 + 2n) + -1 = 19

Multiply (4 + 2n) * (-1 + 2n)
(4(-1 + 2n) + 2n * (-1 + 2n)) + -1 = 19
((-1 * 4 + 2n * 4) + 2n * (-1 + 2n)) + -1 = 19
((-4 + 8n) + 2n * (-1 + 2n)) + -1 = 19
(-4 + 8n + (-1 * 2n + 2n * 2n)) + -1 = 19
(-4 + 8n + (-2n + 4n2)) + -1 = 19

Combine like terms: 8n + -2n = 6n
(-4 + 6n + 4n2) + -1 = 19

Reorder the terms:
-4 + -1 + 6n + 4n2 = 19

Combine like terms: -4 + -1 = -5
-5 + 6n + 4n2 = 19

Solving
-5 + 6n + 4n2 = 19

Solving for variable 'n'.

Reorder the terms:
-5 + -19 + 6n + 4n2 = 19 + -19

Combine like terms: -5 + -19 = -24
-24 + 6n + 4n2 = 19 + -19

Combine like terms: 19 + -19 = 0
-24 + 6n + 4n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-12 + 3n + 2n2) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(-12 + 3n + 2n2)' equal to zero and attempt to solve:

Simplifying
-12 + 3n + 2n2 = 0

Solving
-12 + 3n + 2n2 = 0

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

Divide each side by '2'.
-6 + 1.5n + n2 = 0

Move the constant term to the right:

Add '6' to each side of the equation.
-6 + 1.5n + 6 + n2 = 0 + 6

Reorder the terms:
-6 + 6 + 1.5n + n2 = 0 + 6

Combine like terms: -6 + 6 = 0
0 + 1.5n + n2 = 0 + 6
1.5n + n2 = 0 + 6

Combine like terms: 0 + 6 = 6
1.5n + n2 = 6

The n term is 1.5n.  Take half its coefficient (0.75).
Square it (0.5625) and add it to both sides.

Add '0.5625' to each side of the equation.
1.5n + 0.5625 + n2 = 6 + 0.5625

Reorder the terms:
0.5625 + 1.5n + n2 = 6 + 0.5625

Combine like terms: 6 + 0.5625 = 6.5625
0.5625 + 1.5n + n2 = 6.5625

Factor a perfect square on the left side:
(n + 0.75)(n + 0.75) = 6.5625

Calculate the square root of the right side: 2.561737691

Break this problem into two subproblems by setting 
(n + 0.75) equal to 2.561737691 and -2.561737691.

Subproblem 1
n + 0.75 = 2.561737691

Simplifying
n + 0.75 = 2.561737691

Reorder the terms:
0.75 + n = 2.561737691

Solving
0.75 + n = 2.561737691

Solving for variable 'n'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + n = 2.561737691 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + n = 2.561737691 + -0.75
n = 2.561737691 + -0.75

Combine like terms: 2.561737691 + -0.75 = 1.811737691
n = 1.811737691

Simplifying
n = 1.811737691

Subproblem 2
n + 0.75 = -2.561737691

Simplifying
n + 0.75 = -2.561737691

Reorder the terms:
0.75 + n = -2.561737691

Solving
0.75 + n = -2.561737691

Solving for variable 'n'.

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

Add '-0.75' to each side of the equation.
0.75 + -0.75 + n = -2.561737691 + -0.75

Combine like terms: 0.75 + -0.75 = 0.00
0.00 + n = -2.561737691 + -0.75
n = -2.561737691 + -0.75

Combine like terms: -2.561737691 + -0.75 = -3.311737691
n = -3.311737691

Simplifying
n = -3.311737691

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {1.811737691, -3.311737691}
Solution
n = {1.811737691, -3.311737691}