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

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

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

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

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

Multiply (1 + 2x) * (2 + x)
(1(2 + x) + 2x * (2 + x)) + -1(x + -1)(3x + 1) = 1
((2 * 1 + x * 1) + 2x * (2 + x)) + -1(x + -1)(3x + 1) = 1
((2 + 1x) + 2x * (2 + x)) + -1(x + -1)(3x + 1) = 1
(2 + 1x + (2 * 2x + x * 2x)) + -1(x + -1)(3x + 1) = 1
(2 + 1x + (4x + 2x2)) + -1(x + -1)(3x + 1) = 1

Combine like terms: 1x + 4x = 5x
(2 + 5x + 2x2) + -1(x + -1)(3x + 1) = 1

Reorder the terms:
2 + 5x + 2x2 + -1(-1 + x)(3x + 1) = 1

Reorder the terms:
2 + 5x + 2x2 + -1(-1 + x)(1 + 3x) = 1

Multiply (-1 + x) * (1 + 3x)
2 + 5x + 2x2 + -1(-1(1 + 3x) + x(1 + 3x)) = 1
2 + 5x + 2x2 + -1((1 * -1 + 3x * -1) + x(1 + 3x)) = 1
2 + 5x + 2x2 + -1((-1 + -3x) + x(1 + 3x)) = 1
2 + 5x + 2x2 + -1(-1 + -3x + (1 * x + 3x * x)) = 1
2 + 5x + 2x2 + -1(-1 + -3x + (1x + 3x2)) = 1

Combine like terms: -3x + 1x = -2x
2 + 5x + 2x2 + -1(-1 + -2x + 3x2) = 1
2 + 5x + 2x2 + (-1 * -1 + -2x * -1 + 3x2 * -1) = 1
2 + 5x + 2x2 + (1 + 2x + -3x2) = 1

Reorder the terms:
2 + 1 + 5x + 2x + 2x2 + -3x2 = 1

Combine like terms: 2 + 1 = 3
3 + 5x + 2x + 2x2 + -3x2 = 1

Combine like terms: 5x + 2x = 7x
3 + 7x + 2x2 + -3x2 = 1

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

Solving
3 + 7x + -1x2 = 1

Solving for variable 'x'.

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

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

Combine like terms: 1 + -1 = 0
2 + 7x + -1x2 = 0

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

Divide each side by '-1'.
-2 + -7x + x2 = 0

Move the constant term to the right:

Add '2' to each side of the equation.
-2 + -7x + 2 + x2 = 0 + 2

Reorder the terms:
-2 + 2 + -7x + x2 = 0 + 2

Combine like terms: -2 + 2 = 0
0 + -7x + x2 = 0 + 2
-7x + x2 = 0 + 2

Combine like terms: 0 + 2 = 2
-7x + x2 = 2

The x term is -7x.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7x + 12.25 + x2 = 2 + 12.25

Reorder the terms:
12.25 + -7x + x2 = 2 + 12.25

Combine like terms: 2 + 12.25 = 14.25
12.25 + -7x + x2 = 14.25

Factor a perfect square on the left side:
(x + -3.5)(x + -3.5) = 14.25

Calculate the square root of the right side: 3.774917218

Break this problem into two subproblems by setting 
(x + -3.5) equal to 3.774917218 and -3.774917218.

Subproblem 1
x + -3.5 = 3.774917218

Simplifying
x + -3.5 = 3.774917218

Reorder the terms:
-3.5 + x = 3.774917218

Solving
-3.5 + x = 3.774917218

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = 3.774917218 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = 3.774917218 + 3.5
x = 3.774917218 + 3.5

Combine like terms: 3.774917218 + 3.5 = 7.274917218
x = 7.274917218

Simplifying
x = 7.274917218

Subproblem 2
x + -3.5 = -3.774917218

Simplifying
x + -3.5 = -3.774917218

Reorder the terms:
-3.5 + x = -3.774917218

Solving
-3.5 + x = -3.774917218

Solving for variable 'x'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + x = -3.774917218 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + x = -3.774917218 + 3.5
x = -3.774917218 + 3.5

Combine like terms: -3.774917218 + 3.5 = -0.274917218
x = -0.274917218

Simplifying
x = -0.274917218

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.274917218, -0.274917218}