8x(x+2)+3(x+1)+1=0

Solution for 8x(x+2)+3(x+1)+1=0 equation:

Simplifying
8x(x + 2) + 3(x + 1) + 1 = 0

Reorder the terms:
8x(2 + x) + 3(x + 1) + 1 = 0
(2 * 8x + x * 8x) + 3(x + 1) + 1 = 0
(16x + 8x2) + 3(x + 1) + 1 = 0

Reorder the terms:
16x + 8x2 + 3(1 + x) + 1 = 0
16x + 8x2 + (1 * 3 + x * 3) + 1 = 0
16x + 8x2 + (3 + 3x) + 1 = 0

Reorder the terms:
3 + 1 + 16x + 3x + 8x2 = 0

Combine like terms: 3 + 1 = 4
4 + 16x + 3x + 8x2 = 0

Combine like terms: 16x + 3x = 19x
4 + 19x + 8x2 = 0

Solving
4 + 19x + 8x2 = 0

Solving for variable 'x'.

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

Divide each side by '8'.
0.5 + 2.375x + x2 = 0

Move the constant term to the right:

Add '-0.5' to each side of the equation.
0.5 + 2.375x + -0.5 + x2 = 0 + -0.5

Reorder the terms:
0.5 + -0.5 + 2.375x + x2 = 0 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + 2.375x + x2 = 0 + -0.5
2.375x + x2 = 0 + -0.5

Combine like terms: 0 + -0.5 = -0.5
2.375x + x2 = -0.5

The x term is 2.375x.  Take half its coefficient (1.1875).
Square it (1.41015625) and add it to both sides.

Add '1.41015625' to each side of the equation.
2.375x + 1.41015625 + x2 = -0.5 + 1.41015625

Reorder the terms:
1.41015625 + 2.375x + x2 = -0.5 + 1.41015625

Combine like terms: -0.5 + 1.41015625 = 0.91015625
1.41015625 + 2.375x + x2 = 0.91015625

Factor a perfect square on the left side:
(x + 1.1875)(x + 1.1875) = 0.91015625

Calculate the square root of the right side: 0.954021095

Break this problem into two subproblems by setting 
(x + 1.1875) equal to 0.954021095 and -0.954021095.

Subproblem 1
x + 1.1875 = 0.954021095

Simplifying
x + 1.1875 = 0.954021095

Reorder the terms:
1.1875 + x = 0.954021095

Solving
1.1875 + x = 0.954021095

Solving for variable 'x'.

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

Add '-1.1875' to each side of the equation.
1.1875 + -1.1875 + x = 0.954021095 + -1.1875

Combine like terms: 1.1875 + -1.1875 = 0.0000
0.0000 + x = 0.954021095 + -1.1875
x = 0.954021095 + -1.1875

Combine like terms: 0.954021095 + -1.1875 = -0.233478905
x = -0.233478905

Simplifying
x = -0.233478905

Subproblem 2
x + 1.1875 = -0.954021095

Simplifying
x + 1.1875 = -0.954021095

Reorder the terms:
1.1875 + x = -0.954021095

Solving
1.1875 + x = -0.954021095

Solving for variable 'x'.

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

Add '-1.1875' to each side of the equation.
1.1875 + -1.1875 + x = -0.954021095 + -1.1875

Combine like terms: 1.1875 + -1.1875 = 0.0000
0.0000 + x = -0.954021095 + -1.1875
x = -0.954021095 + -1.1875

Combine like terms: -0.954021095 + -1.1875 = -2.141521095
x = -2.141521095

Simplifying
x = -2.141521095

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.233478905, -2.141521095}