(3x+1)(3x-4)-x(2x+3)=-1

Solution for (3x+1)(3x-4)-x(2x+3)=-1 equation:

Simplifying
(3x + 1)(3x + -4) + -1x(2x + 3) = -1

Reorder the terms:
(1 + 3x)(3x + -4) + -1x(2x + 3) = -1

Reorder the terms:
(1 + 3x)(-4 + 3x) + -1x(2x + 3) = -1

Multiply (1 + 3x) * (-4 + 3x)
(1(-4 + 3x) + 3x * (-4 + 3x)) + -1x(2x + 3) = -1
((-4 * 1 + 3x * 1) + 3x * (-4 + 3x)) + -1x(2x + 3) = -1
((-4 + 3x) + 3x * (-4 + 3x)) + -1x(2x + 3) = -1
(-4 + 3x + (-4 * 3x + 3x * 3x)) + -1x(2x + 3) = -1
(-4 + 3x + (-12x + 9x2)) + -1x(2x + 3) = -1

Combine like terms: 3x + -12x = -9x
(-4 + -9x + 9x2) + -1x(2x + 3) = -1

Reorder the terms:
-4 + -9x + 9x2 + -1x(3 + 2x) = -1
-4 + -9x + 9x2 + (3 * -1x + 2x * -1x) = -1
-4 + -9x + 9x2 + (-3x + -2x2) = -1

Reorder the terms:
-4 + -9x + -3x + 9x2 + -2x2 = -1

Combine like terms: -9x + -3x = -12x
-4 + -12x + 9x2 + -2x2 = -1

Combine like terms: 9x2 + -2x2 = 7x2
-4 + -12x + 7x2 = -1

Solving
-4 + -12x + 7x2 = -1

Solving for variable 'x'.

Reorder the terms:
-4 + 1 + -12x + 7x2 = -1 + 1

Combine like terms: -4 + 1 = -3
-3 + -12x + 7x2 = -1 + 1

Combine like terms: -1 + 1 = 0
-3 + -12x + 7x2 = 0

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

Divide each side by '7'.
-0.4285714286 + -1.714285714x + x2 = 0

Move the constant term to the right:

Add '0.4285714286' to each side of the equation.
-0.4285714286 + -1.714285714x + 0.4285714286 + x2 = 0 + 0.4285714286

Reorder the terms:
-0.4285714286 + 0.4285714286 + -1.714285714x + x2 = 0 + 0.4285714286

Combine like terms: -0.4285714286 + 0.4285714286 = 0.0000000000
0.0000000000 + -1.714285714x + x2 = 0 + 0.4285714286
-1.714285714x + x2 = 0 + 0.4285714286

Combine like terms: 0 + 0.4285714286 = 0.4285714286
-1.714285714x + x2 = 0.4285714286

The x term is -1.714285714x.  Take half its coefficient (-0.857142857).
Square it (0.7346938773) and add it to both sides.

Add '0.7346938773' to each side of the equation.
-1.714285714x + 0.7346938773 + x2 = 0.4285714286 + 0.7346938773

Reorder the terms:
0.7346938773 + -1.714285714x + x2 = 0.4285714286 + 0.7346938773

Combine like terms: 0.4285714286 + 0.7346938773 = 1.1632653059
0.7346938773 + -1.714285714x + x2 = 1.1632653059

Factor a perfect square on the left side:
(x + -0.857142857)(x + -0.857142857) = 1.1632653059

Calculate the square root of the right side: 1.078547776

Break this problem into two subproblems by setting 
(x + -0.857142857) equal to 1.078547776 and -1.078547776.

Subproblem 1
x + -0.857142857 = 1.078547776

Simplifying
x + -0.857142857 = 1.078547776

Reorder the terms:
-0.857142857 + x = 1.078547776

Solving
-0.857142857 + x = 1.078547776

Solving for variable 'x'.

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

Add '0.857142857' to each side of the equation.
-0.857142857 + 0.857142857 + x = 1.078547776 + 0.857142857

Combine like terms: -0.857142857 + 0.857142857 = 0.000000000
0.000000000 + x = 1.078547776 + 0.857142857
x = 1.078547776 + 0.857142857

Combine like terms: 1.078547776 + 0.857142857 = 1.935690633
x = 1.935690633

Simplifying
x = 1.935690633

Subproblem 2
x + -0.857142857 = -1.078547776

Simplifying
x + -0.857142857 = -1.078547776

Reorder the terms:
-0.857142857 + x = -1.078547776

Solving
-0.857142857 + x = -1.078547776

Solving for variable 'x'.

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

Add '0.857142857' to each side of the equation.
-0.857142857 + 0.857142857 + x = -1.078547776 + 0.857142857

Combine like terms: -0.857142857 + 0.857142857 = 0.000000000
0.000000000 + x = -1.078547776 + 0.857142857
x = -1.078547776 + 0.857142857

Combine like terms: -1.078547776 + 0.857142857 = -0.221404919
x = -0.221404919

Simplifying
x = -0.221404919

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.935690633, -0.221404919}