(4n+1)(4n+1)+3(4n+1)-4=0

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Solution for (4n+1)(4n+1)+3(4n+1)-4=0 equation: 


Simplifying
(4n + 1)(4n + 1) + 3(4n + 1) + -4 = 0

Reorder the terms:
(1 + 4n)(4n + 1) + 3(4n + 1) + -4 = 0

Reorder the terms:
(1 + 4n)(1 + 4n) + 3(4n + 1) + -4 = 0

Multiply (1 + 4n) * (1 + 4n)
(1(1 + 4n) + 4n * (1 + 4n)) + 3(4n + 1) + -4 = 0
((1 * 1 + 4n * 1) + 4n * (1 + 4n)) + 3(4n + 1) + -4 = 0
((1 + 4n) + 4n * (1 + 4n)) + 3(4n + 1) + -4 = 0
(1 + 4n + (1 * 4n + 4n * 4n)) + 3(4n + 1) + -4 = 0
(1 + 4n + (4n + 16n2)) + 3(4n + 1) + -4 = 0

Combine like terms: 4n + 4n = 8n
(1 + 8n + 16n2) + 3(4n + 1) + -4 = 0

Reorder the terms:
1 + 8n + 16n2 + 3(1 + 4n) + -4 = 0
1 + 8n + 16n2 + (1 * 3 + 4n * 3) + -4 = 0
1 + 8n + 16n2 + (3 + 12n) + -4 = 0

Reorder the terms:
1 + 3 + -4 + 8n + 12n + 16n2 = 0

Combine like terms: 1 + 3 = 4
4 + -4 + 8n + 12n + 16n2 = 0

Combine like terms: 4 + -4 = 0
0 + 8n + 12n + 16n2 = 0
8n + 12n + 16n2 = 0

Combine like terms: 8n + 12n = 20n
20n + 16n2 = 0

Solving
20n + 16n2 = 0

Solving for variable 'n'.

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

Ignore the factor 4.

Subproblem 1
Set the factor 'n' equal to zero and attempt to solve:

Simplifying
n = 0

Solving
n = 0

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

Simplifying
n = 0

Subproblem 2
Set the factor '(5 + 4n)' equal to zero and attempt to solve:

Simplifying
5 + 4n = 0

Solving
5 + 4n = 0

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

Add '-5' to each side of the equation.
5 + -5 + 4n = 0 + -5

Combine like terms: 5 + -5 = 0
0 + 4n = 0 + -5
4n = 0 + -5

Combine like terms: 0 + -5 = -5
4n = -5

Divide each side by '4'.
n = -1.25

Simplifying
n = -1.25
Solution
n = {0, -1.25}

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