X^2-2x-15/x^2+x-12*2x^2-6x/x^3+3x^2

Brackets are absolutely essential here.

I am sure you meant

(x^2-2x-15)/(x^2+x-12)*(2x^2-6x)/(x^3+3x^2)

= (x-5)(x+3)/( (x+4)(x-3) ) * 2x(x-3)/( x^2(x+3))
= 2(x-5)/(x(x+4)) or (2x -10)/(x^2 + 4x) , x ≠ ±3,0

To simplify the given expression:

x^2 - 2x - 15
---------------- * (2x^2 - 6x)
x^2 + x - 12 x^3 + 3x^2

We'll break it down into smaller parts and simplify separately.

First, let's simplify the numerator.

The expression x^2 - 2x - 15 can be factored as (x - 5)(x + 3).

So, the numerator becomes (x - 5)(x + 3).

Now, let's simplify the denominator.

For the denominator x^2 + x - 12, it can be factored as (x + 4)(x - 3).

The denominator becomes (x + 4)(x - 3).

Next, let's simplify the second part of the expression, (2x^2 - 6x).

We can factor 2x out from each term:

2x(x - 3)

Now, let's simplify the denominator of the second part, x^3 + 3x^2.

We can pull out an x^2 common factor:

x^2(x + 3)

Now, we can cancel out common factors between the numerator and denominator:

[(x - 5)(x + 3)] / [(x + 4)(x - 3)] * [2x(x - 3)] / [x^2(x + 3)]

Canceling common factors gives us:

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

Finally, we can simplify the expression further by multiplying:

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

This can be simplified to:

2x - 10 / x + 4x

or

(2x - 10) / (5x)

That's the simplified form of the given expression.