4+5+6+7+8+4n=2(n+1)(4n-3)

Solution for 4+5+6+7+8+4n=2(n+1)(4n-3) equation:

Simplifying
4 + 5 + 6 + 7 + 8 + 4n = 2(n + 1)(4n + -3)

Combine like terms: 4 + 5 = 9
9 + 6 + 7 + 8 + 4n = 2(n + 1)(4n + -3)

Combine like terms: 9 + 6 = 15
15 + 7 + 8 + 4n = 2(n + 1)(4n + -3)

Combine like terms: 15 + 7 = 22
22 + 8 + 4n = 2(n + 1)(4n + -3)

Combine like terms: 22 + 8 = 30
30 + 4n = 2(n + 1)(4n + -3)

Reorder the terms:
30 + 4n = 2(1 + n)(4n + -3)

Reorder the terms:
30 + 4n = 2(1 + n)(-3 + 4n)

Multiply (1 + n) * (-3 + 4n)
30 + 4n = 2(1(-3 + 4n) + n(-3 + 4n))
30 + 4n = 2((-3 * 1 + 4n * 1) + n(-3 + 4n))
30 + 4n = 2((-3 + 4n) + n(-3 + 4n))
30 + 4n = 2(-3 + 4n + (-3 * n + 4n * n))
30 + 4n = 2(-3 + 4n + (-3n + 4n2))

Combine like terms: 4n + -3n = 1n
30 + 4n = 2(-3 + 1n + 4n2)
30 + 4n = (-3 * 2 + 1n * 2 + 4n2 * 2)
30 + 4n = (-6 + 2n + 8n2)

Solving
30 + 4n = -6 + 2n + 8n2

Solving for variable 'n'.

Reorder the terms:
30 + 6 + 4n + -2n + -8n2 = -6 + 2n + 8n2 + 6 + -2n + -8n2

Combine like terms: 30 + 6 = 36
36 + 4n + -2n + -8n2 = -6 + 2n + 8n2 + 6 + -2n + -8n2

Combine like terms: 4n + -2n = 2n
36 + 2n + -8n2 = -6 + 2n + 8n2 + 6 + -2n + -8n2

Reorder the terms:
36 + 2n + -8n2 = -6 + 6 + 2n + -2n + 8n2 + -8n2

Combine like terms: -6 + 6 = 0
36 + 2n + -8n2 = 0 + 2n + -2n + 8n2 + -8n2
36 + 2n + -8n2 = 2n + -2n + 8n2 + -8n2

Combine like terms: 2n + -2n = 0
36 + 2n + -8n2 = 0 + 8n2 + -8n2
36 + 2n + -8n2 = 8n2 + -8n2

Combine like terms: 8n2 + -8n2 = 0
36 + 2n + -8n2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(18 + n + -4n2) = 0

Factor a trinomial.
2((9 + -4n)(2 + n)) = 0

Ignore the factor 2.
Subproblem 1
Set the factor '(9 + -4n)' equal to zero and attempt to solve:

Simplifying
9 + -4n = 0

Solving
9 + -4n = 0

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

Add '-9' to each side of the equation.
9 + -9 + -4n = 0 + -9

Combine like terms: 9 + -9 = 0
0 + -4n = 0 + -9
-4n = 0 + -9

Combine like terms: 0 + -9 = -9
-4n = -9

Divide each side by '-4'.
n = 2.25

Simplifying
n = 2.25
Subproblem 2
Set the factor '(2 + n)' equal to zero and attempt to solve:

Simplifying
2 + n = 0

Solving
2 + n = 0

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

Add '-2' to each side of the equation.
2 + -2 + n = 0 + -2

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

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

Simplifying
n = -2
Solution
n = {2.25, -2}