(3x+1)(x+2)=-1

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

Simplifying
(3x + 1)(x + 2) = -1

Reorder the terms:
(1 + 3x)(x + 2) = -1

Reorder the terms:
(1 + 3x)(2 + x) = -1

Multiply (1 + 3x) * (2 + x)
(1(2 + x) + 3x * (2 + x)) = -1
((2 * 1 + x * 1) + 3x * (2 + x)) = -1
((2 + 1x) + 3x * (2 + x)) = -1
(2 + 1x + (2 * 3x + x * 3x)) = -1
(2 + 1x + (6x + 3x2)) = -1

Combine like terms: 1x + 6x = 7x
(2 + 7x + 3x2) = -1

Solving
2 + 7x + 3x2 = -1

Solving for variable 'x'.

Reorder the terms:
2 + 1 + 7x + 3x2 = -1 + 1

Combine like terms: 2 + 1 = 3
3 + 7x + 3x2 = -1 + 1

Combine like terms: -1 + 1 = 0
3 + 7x + 3x2 = 0

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

Divide each side by '3'.
1 + 2.333333333x + x2 = 0

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 2.333333333x + -1 + x2 = 0 + -1

Reorder the terms:
1 + -1 + 2.333333333x + x2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 2.333333333x + x2 = 0 + -1
2.333333333x + x2 = 0 + -1

Combine like terms: 0 + -1 = -1
2.333333333x + x2 = -1

The x term is 2.333333333x.  Take half its coefficient (1.166666667).
Square it (1.361111112) and add it to both sides.

Add '1.361111112' to each side of the equation.
2.333333333x + 1.361111112 + x2 = -1 + 1.361111112

Reorder the terms:
1.361111112 + 2.333333333x + x2 = -1 + 1.361111112

Combine like terms: -1 + 1.361111112 = 0.361111112
1.361111112 + 2.333333333x + x2 = 0.361111112

Factor a perfect square on the left side:
(x + 1.166666667)(x + 1.166666667) = 0.361111112

Calculate the square root of the right side: 0.600925213

Break this problem into two subproblems by setting 
(x + 1.166666667) equal to 0.600925213 and -0.600925213.

Subproblem 1
x + 1.166666667 = 0.600925213

Simplifying
x + 1.166666667 = 0.600925213

Reorder the terms:
1.166666667 + x = 0.600925213

Solving
1.166666667 + x = 0.600925213

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = 0.600925213 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = 0.600925213 + -1.166666667
x = 0.600925213 + -1.166666667

Combine like terms: 0.600925213 + -1.166666667 = -0.565741454
x = -0.565741454

Simplifying
x = -0.565741454

Subproblem 2
x + 1.166666667 = -0.600925213

Simplifying
x + 1.166666667 = -0.600925213

Reorder the terms:
1.166666667 + x = -0.600925213

Solving
1.166666667 + x = -0.600925213

Solving for variable 'x'.

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

Add '-1.166666667' to each side of the equation.
1.166666667 + -1.166666667 + x = -0.600925213 + -1.166666667

Combine like terms: 1.166666667 + -1.166666667 = 0.000000000
0.000000000 + x = -0.600925213 + -1.166666667
x = -0.600925213 + -1.166666667

Combine like terms: -0.600925213 + -1.166666667 = -1.76759188
x = -1.76759188

Simplifying
x = -1.76759188

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-0.565741454, -1.76759188}