(5x+4)(x+1)=1

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

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

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

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

Multiply (4 + 5x) * (1 + x)
(4(1 + x) + 5x * (1 + x)) = 1
((1 * 4 + x * 4) + 5x * (1 + x)) = 1
((4 + 4x) + 5x * (1 + x)) = 1
(4 + 4x + (1 * 5x + x * 5x)) = 1
(4 + 4x + (5x + 5x2)) = 1

Combine like terms: 4x + 5x = 9x
(4 + 9x + 5x2) = 1

Solving
4 + 9x + 5x2 = 1

Solving for variable 'x'.

Reorder the terms:
4 + -1 + 9x + 5x2 = 1 + -1

Combine like terms: 4 + -1 = 3
3 + 9x + 5x2 = 1 + -1

Combine like terms: 1 + -1 = 0
3 + 9x + 5x2 = 0

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

Divide each side by '5'.
0.6 + 1.8x + x2 = 0

Move the constant term to the right:

Add '-0.6' to each side of the equation.
0.6 + 1.8x + -0.6 + x2 = 0 + -0.6

Reorder the terms:
0.6 + -0.6 + 1.8x + x2 = 0 + -0.6

Combine like terms: 0.6 + -0.6 = 0.0
0.0 + 1.8x + x2 = 0 + -0.6
1.8x + x2 = 0 + -0.6

Combine like terms: 0 + -0.6 = -0.6
1.8x + x2 = -0.6

The x term is 1.8x.  Take half its coefficient (0.9).
Square it (0.81) and add it to both sides.

Add '0.81' to each side of the equation.
1.8x + 0.81 + x2 = -0.6 + 0.81

Reorder the terms:
0.81 + 1.8x + x2 = -0.6 + 0.81

Combine like terms: -0.6 + 0.81 = 0.21
0.81 + 1.8x + x2 = 0.21

Factor a perfect square on the left side:
(x + 0.9)(x + 0.9) = 0.21

Calculate the square root of the right side: 0.458257569

Break this problem into two subproblems by setting 
(x + 0.9) equal to 0.458257569 and -0.458257569.

Subproblem 1
x + 0.9 = 0.458257569

Simplifying
x + 0.9 = 0.458257569

Reorder the terms:
0.9 + x = 0.458257569

Solving
0.9 + x = 0.458257569

Solving for variable 'x'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + x = 0.458257569 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + x = 0.458257569 + -0.9
x = 0.458257569 + -0.9

Combine like terms: 0.458257569 + -0.9 = -0.441742431
x = -0.441742431

Simplifying
x = -0.441742431

Subproblem 2
x + 0.9 = -0.458257569

Simplifying
x + 0.9 = -0.458257569

Reorder the terms:
0.9 + x = -0.458257569

Solving
0.9 + x = -0.458257569

Solving for variable 'x'.

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

Add '-0.9' to each side of the equation.
0.9 + -0.9 + x = -0.458257569 + -0.9

Combine like terms: 0.9 + -0.9 = 0.0
0.0 + x = -0.458257569 + -0.9
x = -0.458257569 + -0.9

Combine like terms: -0.458257569 + -0.9 = -1.358257569
x = -1.358257569

Simplifying
x = -1.358257569

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.441742431, -1.358257569}