3x(x^2-4x)+(x-2)(x^2+3x)=0

Solution for 3x(x^2-4x)+(x-2)(x^2+3x)=0 equation:

Simplifying
3x(x2 + -4x) + (x + -2)(x2 + 3x) = 0

Reorder the terms:
3x(-4x + x2) + (x + -2)(x2 + 3x) = 0
(-4x * 3x + x2 * 3x) + (x + -2)(x2 + 3x) = 0
(-12x2 + 3x3) + (x + -2)(x2 + 3x) = 0

Reorder the terms:
-12x2 + 3x3 + (-2 + x)(x2 + 3x) = 0

Reorder the terms:
-12x2 + 3x3 + (-2 + x)(3x + x2) = 0

Multiply (-2 + x) * (3x + x2)
-12x2 + 3x3 + (-2(3x + x2) + x(3x + x2)) = 0
-12x2 + 3x3 + ((3x * -2 + x2 * -2) + x(3x + x2)) = 0
-12x2 + 3x3 + ((-6x + -2x2) + x(3x + x2)) = 0
-12x2 + 3x3 + (-6x + -2x2 + (3x * x + x2 * x)) = 0
-12x2 + 3x3 + (-6x + -2x2 + (3x2 + x3)) = 0

Combine like terms: -2x2 + 3x2 = 1x2
-12x2 + 3x3 + (-6x + 1x2 + x3) = 0

Reorder the terms:
-6x + -12x2 + 1x2 + 3x3 + x3 = 0

Combine like terms: -12x2 + 1x2 = -11x2
-6x + -11x2 + 3x3 + x3 = 0

Combine like terms: 3x3 + x3 = 4x3
-6x + -11x2 + 4x3 = 0

Solving
-6x + -11x2 + 4x3 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), 'x'.
x(-6 + -11x + 4x2) = 0

Subproblem 1
Set the factor 'x' equal to zero and attempt to solve:

Simplifying
x = 0

Solving
x = 0

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

Simplifying
x = 0
Subproblem 2
Set the factor '(-6 + -11x + 4x2)' equal to zero and attempt to solve:

Simplifying
-6 + -11x + 4x2 = 0

Solving
-6 + -11x + 4x2 = 0

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

Divide each side by '4'.
-1.5 + -2.75x + x2 = 0

Move the constant term to the right:

Add '1.5' to each side of the equation.
-1.5 + -2.75x + 1.5 + x2 = 0 + 1.5

Reorder the terms:
-1.5 + 1.5 + -2.75x + x2 = 0 + 1.5

Combine like terms: -1.5 + 1.5 = 0.0
0.0 + -2.75x + x2 = 0 + 1.5
-2.75x + x2 = 0 + 1.5

Combine like terms: 0 + 1.5 = 1.5
-2.75x + x2 = 1.5

The x term is -2.75x.  Take half its coefficient (-1.375).
Square it (1.890625) and add it to both sides.

Add '1.890625' to each side of the equation.
-2.75x + 1.890625 + x2 = 1.5 + 1.890625

Reorder the terms:
1.890625 + -2.75x + x2 = 1.5 + 1.890625

Combine like terms: 1.5 + 1.890625 = 3.390625
1.890625 + -2.75x + x2 = 3.390625

Factor a perfect square on the left side:
(x + -1.375)(x + -1.375) = 3.390625

Calculate the square root of the right side: 1.841364983

Break this problem into two subproblems by setting 
(x + -1.375) equal to 1.841364983 and -1.841364983.

Subproblem 1
x + -1.375 = 1.841364983

Simplifying
x + -1.375 = 1.841364983

Reorder the terms:
-1.375 + x = 1.841364983

Solving
-1.375 + x = 1.841364983

Solving for variable 'x'.

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

Add '1.375' to each side of the equation.
-1.375 + 1.375 + x = 1.841364983 + 1.375

Combine like terms: -1.375 + 1.375 = 0.000
0.000 + x = 1.841364983 + 1.375
x = 1.841364983 + 1.375

Combine like terms: 1.841364983 + 1.375 = 3.216364983
x = 3.216364983

Simplifying
x = 3.216364983

Subproblem 2
x + -1.375 = -1.841364983

Simplifying
x + -1.375 = -1.841364983

Reorder the terms:
-1.375 + x = -1.841364983

Solving
-1.375 + x = -1.841364983

Solving for variable 'x'.

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

Add '1.375' to each side of the equation.
-1.375 + 1.375 + x = -1.841364983 + 1.375

Combine like terms: -1.375 + 1.375 = 0.000
0.000 + x = -1.841364983 + 1.375
x = -1.841364983 + 1.375

Combine like terms: -1.841364983 + 1.375 = -0.466364983
x = -0.466364983

Simplifying
x = -0.466364983

Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.216364983, -0.466364983}
Solution
x = {0, 3.216364983, -0.466364983}