4k(4)+32k(3)+60k(2)=4k(2)(k+3)(k+5)

Solution for 4k(4)+32k(3)+60k(2)=4k(2)(k+3)(k+5) equation:

Simplifying
4k(4) + 32k(3) + 60k(2) = 4k(2)(k + 3)(k + 5)

Reorder the terms for easier multiplication:
4 * 4k + 32k(3) + 60k(2) = 4k(2)(k + 3)(k + 5)

Multiply 4 * 4
16k + 32k(3) + 60k(2) = 4k(2)(k + 3)(k + 5)

Reorder the terms for easier multiplication:
16k + 32 * 3k + 60k(2) = 4k(2)(k + 3)(k + 5)

Multiply 32 * 3
16k + 96k + 60k(2) = 4k(2)(k + 3)(k + 5)

Reorder the terms for easier multiplication:
16k + 96k + 60 * 2k = 4k(2)(k + 3)(k + 5)

Multiply 60 * 2
16k + 96k + 120k = 4k(2)(k + 3)(k + 5)

Combine like terms: 16k + 96k = 112k
112k + 120k = 4k(2)(k + 3)(k + 5)

Combine like terms: 112k + 120k = 232k
232k = 4k(2)(k + 3)(k + 5)

Reorder the terms:
232k = 4k * 2(3 + k)(k + 5)

Reorder the terms:
232k = 4k * 2(3 + k)(5 + k)

Reorder the terms for easier multiplication:
232k = 4 * 2k(3 + k)(5 + k)

Multiply 4 * 2
232k = 8k(3 + k)(5 + k)

Multiply (3 + k) * (5 + k)
232k = 8k(3(5 + k) + k(5 + k))
232k = 8k((5 * 3 + k * 3) + k(5 + k))
232k = 8k((15 + 3k) + k(5 + k))
232k = 8k(15 + 3k + (5 * k + k * k))
232k = 8k(15 + 3k + (5k + k2))

Combine like terms: 3k + 5k = 8k
232k = 8k(15 + 8k + k2)
232k = (15 * 8k + 8k * 8k + k2 * 8k)
232k = (120k + 64k2 + 8k3)

Solving
232k = 120k + 64k2 + 8k3

Solving for variable 'k'.

Combine like terms: 232k + -120k = 112k
112k + -64k2 + -8k3 = 120k + 64k2 + 8k3 + -120k + -64k2 + -8k3

Reorder the terms:
112k + -64k2 + -8k3 = 120k + -120k + 64k2 + -64k2 + 8k3 + -8k3

Combine like terms: 120k + -120k = 0
112k + -64k2 + -8k3 = 0 + 64k2 + -64k2 + 8k3 + -8k3
112k + -64k2 + -8k3 = 64k2 + -64k2 + 8k3 + -8k3

Combine like terms: 64k2 + -64k2 = 0
112k + -64k2 + -8k3 = 0 + 8k3 + -8k3
112k + -64k2 + -8k3 = 8k3 + -8k3

Combine like terms: 8k3 + -8k3 = 0
112k + -64k2 + -8k3 = 0

Factor out the Greatest Common Factor (GCF), '8k'.
8k(14 + -8k + -1k2) = 0

Ignore the factor 8.
Subproblem 1
Set the factor 'k' equal to zero and attempt to solve:

Simplifying
k = 0

Solving
k = 0

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

Simplifying
k = 0
Subproblem 2
Set the factor '(14 + -8k + -1k2)' equal to zero and attempt to solve:

Simplifying
14 + -8k + -1k2 = 0

Solving
14 + -8k + -1k2 = 0

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
-14 + 8k + k2 = 0

Move the constant term to the right:

Add '14' to each side of the equation.
-14 + 8k + 14 + k2 = 0 + 14

Reorder the terms:
-14 + 14 + 8k + k2 = 0 + 14

Combine like terms: -14 + 14 = 0
0 + 8k + k2 = 0 + 14
8k + k2 = 0 + 14

Combine like terms: 0 + 14 = 14
8k + k2 = 14

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

Add '16' to each side of the equation.
8k + 16 + k2 = 14 + 16

Reorder the terms:
16 + 8k + k2 = 14 + 16

Combine like terms: 14 + 16 = 30
16 + 8k + k2 = 30

Factor a perfect square on the left side:
(k + 4)(k + 4) = 30

Calculate the square root of the right side: 5.477225575

Break this problem into two subproblems by setting 
(k + 4) equal to 5.477225575 and -5.477225575.

Subproblem 1
k + 4 = 5.477225575

Simplifying
k + 4 = 5.477225575

Reorder the terms:
4 + k = 5.477225575

Solving
4 + k = 5.477225575

Solving for variable 'k'.

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

Add '-4' to each side of the equation.
4 + -4 + k = 5.477225575 + -4

Combine like terms: 4 + -4 = 0
0 + k = 5.477225575 + -4
k = 5.477225575 + -4

Combine like terms: 5.477225575 + -4 = 1.477225575
k = 1.477225575

Simplifying
k = 1.477225575

Subproblem 2
k + 4 = -5.477225575

Simplifying
k + 4 = -5.477225575

Reorder the terms:
4 + k = -5.477225575

Solving
4 + k = -5.477225575

Solving for variable 'k'.

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

Add '-4' to each side of the equation.
4 + -4 + k = -5.477225575 + -4

Combine like terms: 4 + -4 = 0
0 + k = -5.477225575 + -4
k = -5.477225575 + -4

Combine like terms: -5.477225575 + -4 = -9.477225575
k = -9.477225575

Simplifying
k = -9.477225575

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {1.477225575, -9.477225575}
Solution
k = {0, 1.477225575, -9.477225575}