3n(2n+5)=7(9n+5)+4

Solution for 3n(2n+5)=7(9n+5)+4 equation:

Simplifying
3n(2n + 5) = 7(9n + 5) + 4

Reorder the terms:
3n(5 + 2n) = 7(9n + 5) + 4
(5 * 3n + 2n * 3n) = 7(9n + 5) + 4
(15n + 6n2) = 7(9n + 5) + 4

Reorder the terms:
15n + 6n2 = 7(5 + 9n) + 4
15n + 6n2 = (5 * 7 + 9n * 7) + 4
15n + 6n2 = (35 + 63n) + 4

Reorder the terms:
15n + 6n2 = 35 + 4 + 63n

Combine like terms: 35 + 4 = 39
15n + 6n2 = 39 + 63n

Solving
15n + 6n2 = 39 + 63n

Solving for variable 'n'.

Reorder the terms:
-39 + 15n + -63n + 6n2 = 39 + 63n + -39 + -63n

Combine like terms: 15n + -63n = -48n
-39 + -48n + 6n2 = 39 + 63n + -39 + -63n

Reorder the terms:
-39 + -48n + 6n2 = 39 + -39 + 63n + -63n

Combine like terms: 39 + -39 = 0
-39 + -48n + 6n2 = 0 + 63n + -63n
-39 + -48n + 6n2 = 63n + -63n

Combine like terms: 63n + -63n = 0
-39 + -48n + 6n2 = 0

Factor out the Greatest Common Factor (GCF), '3'.
3(-13 + -16n + 2n2) = 0

Ignore the factor 3.
Subproblem 1
Set the factor '(-13 + -16n + 2n2)' equal to zero and attempt to solve:

Simplifying
-13 + -16n + 2n2 = 0

Solving
-13 + -16n + 2n2 = 0

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

Divide each side by '2'.
-6.5 + -8n + n2 = 0

Move the constant term to the right:

Add '6.5' to each side of the equation.
-6.5 + -8n + 6.5 + n2 = 0 + 6.5

Reorder the terms:
-6.5 + 6.5 + -8n + n2 = 0 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + -8n + n2 = 0 + 6.5
-8n + n2 = 0 + 6.5

Combine like terms: 0 + 6.5 = 6.5
-8n + n2 = 6.5

The n term is -8n.  Take half its coefficient (-4).
Square it (16) and add it to both sides.

Add '16' to each side of the equation.
-8n + 16 + n2 = 6.5 + 16

Reorder the terms:
16 + -8n + n2 = 6.5 + 16

Combine like terms: 6.5 + 16 = 22.5
16 + -8n + n2 = 22.5

Factor a perfect square on the left side:
(n + -4)(n + -4) = 22.5

Calculate the square root of the right side: 4.74341649

Break this problem into two subproblems by setting 
(n + -4) equal to 4.74341649 and -4.74341649.

Subproblem 1
n + -4 = 4.74341649

Simplifying
n + -4 = 4.74341649

Reorder the terms:
-4 + n = 4.74341649

Solving
-4 + n = 4.74341649

Solving for variable 'n'.

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

Add '4' to each side of the equation.
-4 + 4 + n = 4.74341649 + 4

Combine like terms: -4 + 4 = 0
0 + n = 4.74341649 + 4
n = 4.74341649 + 4

Combine like terms: 4.74341649 + 4 = 8.74341649
n = 8.74341649

Simplifying
n = 8.74341649

Subproblem 2
n + -4 = -4.74341649

Simplifying
n + -4 = -4.74341649

Reorder the terms:
-4 + n = -4.74341649

Solving
-4 + n = -4.74341649

Solving for variable 'n'.

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

Add '4' to each side of the equation.
-4 + 4 + n = -4.74341649 + 4

Combine like terms: -4 + 4 = 0
0 + n = -4.74341649 + 4
n = -4.74341649 + 4

Combine like terms: -4.74341649 + 4 = -0.74341649
n = -0.74341649

Simplifying
n = -0.74341649

Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {8.74341649, -0.74341649}
Solution
n = {8.74341649, -0.74341649}