Simply this rational expression and show your steps: 5x/(4x^(3)-4x^(2)-24x) + 2/(4x^(2)-12x) - 3x/(x^(2)+2x).?

Wow, you are going crazy with the parentheses. I will clean that up a bit so I can see what's what.

5x/(4x^3 - 4x^2 - 24x) + 2/(4x^2 - 12x) - 3x/(x^2 + 2x)

Ok, that's not so bad. It's busy work, but that's ok, you need the practice. Let's start by checking factors.

5x/(4x(x^2 - x - 6)) + 2/(4x(x - 3)) - 3x/(x(x + 2))

5/(4(x - 3)(x + 2)) + 1/(2x(x - 3)) - 3/(x + 2) Now multiply by ((x - 3)(x + 2))/((x - 3)(x + 2)).

5/4 + (x + 2)/(2x) - 3(x - 3)

Always remember that when you multiply or divide by the variable, it is possible to create false results or conceal valid results. So all answers have to be checked in the original statement. For example, you have undefined states at x=-2 and x=3 that are not found in this result.

Factor the denominators:

4x(x - x - 6) = 4x(x + 2)(x - 3)

4x(x - 3)

x(x + 2) so the LCD is 4x(x + 2)(x - 3)

The first expression already has that so leave it as is. The second is missing the (x+2) so multiply top and bottom by (x + 2) making the top be 2(x + 2) = 2x + 4.

The third is missing the 4 and the (x - 3) so multiply top and bottom by 4(x - 3)

making the top be 12x(x - 3) = 12x^2 - 36x.

Now just combine the tops: 5x + 2x + 4 - (12x^2 - 36x)

= 7x + 4 - 12x^2 + 36x

= -12x^2 + 43x + 4; since this does not factor. this over the LCD is the simplest form