(4k+5)(k+1)=1

Solution for (4k+5)(k+1)=1 equation:

Simplifying
(4k + 5)(k + 1) = 1

Reorder the terms:
(5 + 4k)(k + 1) = 1

Reorder the terms:
(5 + 4k)(1 + k) = 1

Multiply (5 + 4k) * (1 + k)
(5(1 + k) + 4k * (1 + k)) = 1
((1 * 5 + k * 5) + 4k * (1 + k)) = 1
((5 + 5k) + 4k * (1 + k)) = 1
(5 + 5k + (1 * 4k + k * 4k)) = 1
(5 + 5k + (4k + 4k2)) = 1

Combine like terms: 5k + 4k = 9k
(5 + 9k + 4k2) = 1

Solving
5 + 9k + 4k2 = 1

Solving for variable 'k'.

Reorder the terms:
5 + -1 + 9k + 4k2 = 1 + -1

Combine like terms: 5 + -1 = 4
4 + 9k + 4k2 = 1 + -1

Combine like terms: 1 + -1 = 0
4 + 9k + 4k2 = 0

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

Divide each side by '4'.
1 + 2.25k + k2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.25k + -1 + k2 = 0 + -1

Reorder the terms:
1 + -1 + 2.25k + k2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.25k + k2 = 0 + -1
2.25k + k2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.25k + k2 = -1

The k term is 2.25k.  Take half its coefficient (1.125).
Square it (1.265625) and add it to both sides.

Add '1.265625' to each side of the equation.
2.25k + 1.265625 + k2 = -1 + 1.265625

Reorder the terms:
1.265625 + 2.25k + k2 = -1 + 1.265625

Combine like terms: -1 + 1.265625 = 0.265625
1.265625 + 2.25k + k2 = 0.265625

Factor a perfect square on the left side:
(k + 1.125)(k + 1.125) = 0.265625

Calculate the square root of the right side: 0.515388203

Break this problem into two subproblems by setting 
(k + 1.125) equal to 0.515388203 and -0.515388203.

Subproblem 1
k + 1.125 = 0.515388203

Simplifying
k + 1.125 = 0.515388203

Reorder the terms:
1.125 + k = 0.515388203

Solving
1.125 + k = 0.515388203

Solving for variable 'k'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + k = 0.515388203 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + k = 0.515388203 + -1.125
k = 0.515388203 + -1.125

Combine like terms: 0.515388203 + -1.125 = -0.609611797
k = -0.609611797

Simplifying
k = -0.609611797

Subproblem 2
k + 1.125 = -0.515388203

Simplifying
k + 1.125 = -0.515388203

Reorder the terms:
1.125 + k = -0.515388203

Solving
1.125 + k = -0.515388203

Solving for variable 'k'.

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

Add '-1.125' to each side of the equation.
1.125 + -1.125 + k = -0.515388203 + -1.125

Combine like terms: 1.125 + -1.125 = 0.000
0.000 + k = -0.515388203 + -1.125
k = -0.515388203 + -1.125

Combine like terms: -0.515388203 + -1.125 = -1.640388203
k = -1.640388203

Simplifying
k = -1.640388203

Solution
The solution to the problem is based on the solutions
from the subproblems.
k = {-0.609611797, -1.640388203}