(3p+n)[(6p+2n)-(5p-4n)]=

Solution for (3p+n)[(6p+2n)-(5p-4n)]= equation:

Simplifying
(3p + n)[(6p + 2n) + -1(5p + -4n)] = 0

Reorder the terms:
(n + 3p)[(6p + 2n) + -1(5p + -4n)] = 0

Reorder the terms:
(n + 3p)[(2n + 6p) + -1(5p + -4n)] = 0

Remove parenthesis around (2n + 6p)
(n + 3p)[2n + 6p + -1(5p + -4n)] = 0

Reorder the terms:
(n + 3p)[2n + 6p + -1(-4n + 5p)] = 0
(n + 3p)[2n + 6p + (-4n * -1 + 5p * -1)] = 0
(n + 3p)[2n + 6p + (4n + -5p)] = 0

Reorder the terms:
(n + 3p)[2n + 4n + 6p + -5p] = 0

Combine like terms: 2n + 4n = 6n
(n + 3p)[6n + 6p + -5p] = 0

Combine like terms: 6p + -5p = 1p
(n + 3p)[6n + 1p] = 0

Multiply (n + 3p) * [6n + 1p]
(n[6n + 1p] + 3p * [6n + 1p]) = 0
([6n * n + 1p * n] + 3p * [6n + 1p]) = 0

Reorder the terms:
([1np + 6n2] + 3p * [6n + 1p]) = 0
([1np + 6n2] + 3p * [6n + 1p]) = 0
(1np + 6n2 + [6n * 3p + 1p * 3p]) = 0
(1np + 6n2 + [18np + 3p2]) = 0

Reorder the terms:
(1np + 18np + 6n2 + 3p2) = 0

Combine like terms: 1np + 18np = 19np
(19np + 6n2 + 3p2) = 0

Solving
19np + 6n2 + 3p2 = 0

Solving for variable 'n'.

Factor a trinomial.
(6n + p)(n + 3p) = 0

Subproblem 1
Set the factor '(6n + p)' equal to zero and attempt to solve:

Simplifying
6n + p = 0

Solving
6n + p = 0

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

Add '-1p' to each side of the equation.
6n + p + -1p = 0 + -1p

Combine like terms: p + -1p = 0
6n + 0 = 0 + -1p
6n = 0 + -1p
Remove the zero:
6n = -1p

Divide each side by '6'.
n = -0.1666666667p

Simplifying
n = -0.1666666667p
Subproblem 2
Set the factor '(n + 3p)' equal to zero and attempt to solve:

Simplifying
n + 3p = 0

Solving
n + 3p = 0

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

Add '-3p' to each side of the equation.
n + 3p + -3p = 0 + -3p

Combine like terms: 3p + -3p = 0
n + 0 = 0 + -3p
n = 0 + -3p
Remove the zero:
n = -3p

Simplifying
n = -3p
Solution
n = {-0.1666666667p, -3p}