x+3 over x cubed - x squared -6x divided by x squared -9 over x squared + x -12.

i got x + 4 over (x squared + 2x)(x-3).
am i right?

you are right!

thanks

To determine if your answer is correct, we can simplify both the numerator and denominator separately and then divide them.

Let's start by simplifying the numerator:

The numerator is (x + 3)/ (x^3 - x^2 - 6x).

We can factor out an x from the denominator and rewrite it as:

(x + 3)/ (x(x^2 - x - 6)).

Factorizing the quadratic term (x^2 - x - 6), we get (x - 3)(x + 2), so the denominator becomes:

(x + 3)/ (x(x - 3)(x + 2)).

Next, let's simplify the denominator:

The denominator is (x^2 - 9)/ (x^2 + x - 12).

We can factorize (x^2 - 9) as (x - 3)(x + 3), so the denominator becomes:

(x - 3)(x + 3)/ (x^2 + x - 12).

Now, when we divide the numerators and denominators, we invert the second fraction and multiply:

[(x + 3)/ (x(x - 3)(x + 2))] * [(x^2 + x - 12)/ (x - 3)(x + 3)]

Note that (x - 3) and (x + 3) cancel out in both the numerator and the denominator:

x + 3/ (x(x + 2)) * (x^2 + x - 12)/ 1

Simplifying further:

(x + 3)(x^2 + x - 12)/ x(x + 2)

Now, let's multiply the terms in the numerator:

x^3 + x^2 - 12x + 3x^2 + 3x - 36/ x(x + 2)

Combining like terms:

x^3 + 4x^2 - 9x - 36/ x(x + 2)

Hence, your answer should be x^3 + 4x^2 - 9x - 36/ x(x + 2), not x + 4/ (x^2 + 2x)(x - 3).