(2k+6)(2k+6)-4(2k+1)(k+5)=0

Solution for (2k+6)(2k+6)-4(2k+1)(k+5)=0 equation:

Simplifying
(2k + 6)(2k + 6) + -4(2k + 1)(k + 5) = 0

Reorder the terms:
(6 + 2k)(2k + 6) + -4(2k + 1)(k + 5) = 0

Reorder the terms:
(6 + 2k)(6 + 2k) + -4(2k + 1)(k + 5) = 0

Multiply (6 + 2k) * (6 + 2k)
(6(6 + 2k) + 2k * (6 + 2k)) + -4(2k + 1)(k + 5) = 0
((6 * 6 + 2k * 6) + 2k * (6 + 2k)) + -4(2k + 1)(k + 5) = 0
((36 + 12k) + 2k * (6 + 2k)) + -4(2k + 1)(k + 5) = 0
(36 + 12k + (6 * 2k + 2k * 2k)) + -4(2k + 1)(k + 5) = 0
(36 + 12k + (12k + 4k2)) + -4(2k + 1)(k + 5) = 0

Combine like terms: 12k + 12k = 24k
(36 + 24k + 4k2) + -4(2k + 1)(k + 5) = 0

Reorder the terms:
36 + 24k + 4k2 + -4(1 + 2k)(k + 5) = 0

Reorder the terms:
36 + 24k + 4k2 + -4(1 + 2k)(5 + k) = 0

Multiply (1 + 2k) * (5 + k)
36 + 24k + 4k2 + -4(1(5 + k) + 2k * (5 + k)) = 0
36 + 24k + 4k2 + -4((5 * 1 + k * 1) + 2k * (5 + k)) = 0
36 + 24k + 4k2 + -4((5 + 1k) + 2k * (5 + k)) = 0
36 + 24k + 4k2 + -4(5 + 1k + (5 * 2k + k * 2k)) = 0
36 + 24k + 4k2 + -4(5 + 1k + (10k + 2k2)) = 0

Combine like terms: 1k + 10k = 11k
36 + 24k + 4k2 + -4(5 + 11k + 2k2) = 0
36 + 24k + 4k2 + (5 * -4 + 11k * -4 + 2k2 * -4) = 0
36 + 24k + 4k2 + (-20 + -44k + -8k2) = 0

Reorder the terms:
36 + -20 + 24k + -44k + 4k2 + -8k2 = 0

Combine like terms: 36 + -20 = 16
16 + 24k + -44k + 4k2 + -8k2 = 0

Combine like terms: 24k + -44k = -20k
16 + -20k + 4k2 + -8k2 = 0

Combine like terms: 4k2 + -8k2 = -4k2
16 + -20k + -4k2 = 0

Solving
16 + -20k + -4k2 = 0

Solving for variable 'k'.

Factor out the Greatest Common Factor (GCF), '4'.
4(4 + -5k + -1k2) = 0

Ignore the factor 4.
Subproblem 1
Set the factor '(4 + -5k + -1k2)' equal to zero and attempt to solve:

Simplifying
4 + -5k + -1k2 = 0

Solving
4 + -5k + -1k2 = 0

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

Divide each side by '-1'.
-4 + 5k + k2 = 0

Move the constant term to the right:

Add '4' to each side of the equation.
-4 + 5k + 4 + k2 = 0 + 4

Reorder the terms:
-4 + 4 + 5k + k2 = 0 + 4

Combine like terms: -4 + 4 = 0
0 + 5k + k2 = 0 + 4
5k + k2 = 0 + 4

Combine like terms: 0 + 4 = 4
5k + k2 = 4

The k term is 5k.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5k + 6.25 + k2 = 4 + 6.25

Reorder the terms:
6.25 + 5k + k2 = 4 + 6.25

Combine like terms: 4 + 6.25 = 10.25
6.25 + 5k + k2 = 10.25

Factor a perfect square on the left side:
(k + 2.5)(k + 2.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(k + 2.5) equal to 3.201562119 and -3.201562119.

Subproblem 1
k + 2.5 = 3.201562119

Simplifying
k + 2.5 = 3.201562119

Reorder the terms:
2.5 + k = 3.201562119

Solving
2.5 + k = 3.201562119

Solving for variable 'k'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + k = 3.201562119 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + k = 3.201562119 + -2.5
k = 3.201562119 + -2.5

Combine like terms: 3.201562119 + -2.5 = 0.701562119
k = 0.701562119

Simplifying
k = 0.701562119

Subproblem 2
k + 2.5 = -3.201562119

Simplifying
k + 2.5 = -3.201562119

Reorder the terms:
2.5 + k = -3.201562119

Solving
2.5 + k = -3.201562119

Solving for variable 'k'.

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

Add '-2.5' to each side of the equation.
2.5 + -2.5 + k = -3.201562119 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + k = -3.201562119 + -2.5
k = -3.201562119 + -2.5

Combine like terms: -3.201562119 + -2.5 = -5.701562119
k = -5.701562119

Simplifying
k = -5.701562119

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {0.701562119, -5.701562119}
Solution
k = {0.701562119, -5.701562119}