Did I do this problem the correct way if so is there a shorter way to explain this?

x(x + 6)(x + -4) = 24x
x(6 + x)(x + -4) = 24x
x(6 + x)(-4 + x) = 24x
(6 + x) * (-4 + x
x(6(-4 + x) + x(-4 + x)) = 24x
x((-4 * 6 + x * 6) + x(-4 + x)) = 24x
x((-24 + 6x) + x(-4 + x)) = 24x
x(-24 + 6x + (-4 * x + x * x)) = 24x
x(-24 + 6x + (-4x + x2)) = 24x
6x + -4x = 2x
x(-24 + 2x + x2) = 24x
(-24 * x + 2x * x + x2 * x) = 24x
(-24x + 2x2 + x3) = 24x
-24x + 2x2 + x3 = 24x
-24x + -24x + 2x2 + x3 = 24x + -24x
-24x + -24x = -48x
-48x + 2x2 + x3 = 24x + -24x
24x + -24x = 0
-48x + 2x2 + x3 = 0
x(-48 + 2x + x2) = 0
x((-8 + -1x)(6 + -1x)) = 0
x = 0
-8 + 8 + -1x = 0 + 8
0 + -1x = 0 + 8
-1x = 0 + 8
-1x = 8
x = -8
x = {0, -8, 6}

First of all, x=0 is clearly a solution.

For other solutions, divide both sides by x to get the quadratic
(x+6)(x-4) = 24
x^2 +2x -24 = 24
x^2 +2x -48 = 0
(x+8)(x-6) = 0
x = -8 or 6

Your answer is correct but you got there the hard way.

It appears that you have correctly solved the problem and obtained the values x = {0, -8, 6} as the solutions. However, the method you have used involves expanding and simplifying the expression step by step. While this is a valid approach, it can be time-consuming and may lead to more opportunities for mistakes.

A shorter and more efficient way to solve this problem is by using the zero product property. The zero product property states that if AB = 0, then either A = 0 or B = 0.

In your equation x(x + 6)(x - 4) = 24x, we can start by setting each factor equal to zero:

Case 1: x = 0
When x = 0, the equation becomes 0(0 + 6)(0 - 4) = 24(0), which simplifies to 0 = 0. This is a true statement, so x = 0 is one solution.

Case 2: x + 6 = 0
Solving for x gives x = -6. Substituting x = -6 back into the equation, we get (-6)(-6 + 6)(-6 - 4) = 24(-6), which simplifies to 0 = 0. Again, this is a true statement, so x = -6 is another solution.

Case 3: x - 4 = 0
Solving for x gives x = 4. Substituting x = 4 back into the equation, we get (4)(4 + 6)(4 - 4) = 24(4), which simplifies to 0 = 0. This is once again a true statement, so x = 4 is a solution as well.

Hence, the solutions to the equation x(x + 6)(x - 4) = 24x are x = {0, -6, 4}.

Using the zero product property allows us to identify the solutions more efficiently compared to expanding and simplifying the expression step by step.