1. [12/(x^2 - 4)] - [(3-x)/(x^2+2x)]

The answer is:
[(x^2 + 7x + 6)/x(x+z)(x-z)
I don't understand how to get that! was the z a typo? even if it's supposed to be a 2 - I don't understand!

yes, the z is a typo

Factor 1/(x+2) out of each term and see what you get.
[1/(x+2)][12/(x-2) - (3-x)/x]
=[1/(x+2)][12x/x(x-2) - (3-x)(x-2)/x(x-2)]
={1/[(x-2)(x+2)]}[12x/x -(3-x)(x-2)/x]
={1/[(x-2)(x+2)]}[(x^2+ 7x +6)/x]
That is in the form of your answer (except for the typo "z" for 2), but it can be factored further.
= (x+6)(x+1)/[x(x-2)(x+2)]

To simplify the given expression, we first need to find a common denominator for both fractions. The denominators of the two fractions are (x^2 - 4) and (x^2 + 2x). The common denominator will be the product of these two denominators:

Common denominator = (x^2 - 4) * (x^2 + 2x)

Next, we need to rewrite the fractions with the common denominator. For the first fraction, we need to multiply the numerator and denominator by (x^2 + 2x) to get a common denominator.
So, the first fraction becomes:

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

For the second fraction, we need to multiply the numerator and denominator by (x^2 - 4) to get a common denominator.
So, the second fraction becomes:

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

Now, we can combine the two fractions over the common denominator:

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

Let's simplify the numerator using the distributive property:

[12x^2 + 24x - (3x^3 - 4x - 12 + 4x^2)] / [(x^2 - 4) * (x^2 + 2x)]

Now, simplify the numerator by combining like terms:

[12x^2 + 24x - 3x^3 + 4x - 12 - 4x^2] / [(x^2 - 4) * (x^2 + 2x)]

Rearranging the terms:

[-3x^3 + (12x^2 - 4x^2) + (24x + 4x) - 12] / [(x^2 - 4) * (x^2 + 2x)]

Simplifying further:

[-3x^3 + 8x^2 + 28x - 12] / [(x^2 - 4) * (x^2 + 2x)]

Now, we can factor out the common factor from the numerator, which is -1:

[-1 * (3x^3 - 8x^2 - 28x + 12)] / [(x^2 - 4) * (x^2 + 2x)]

Factoring the numerator:

[-1 * (3x^3 - 2x^2 - 6x^2 + 4x - 7x + 12)] / [(x^2 - 4) * (x^2 + 2x)]

Grouping the terms:

[-1 * (x^2(3x - 2) - 2x(3x - 2) - 7(3x - 2))] / [(x^2 - 4) * (x^2 + 2x)]

Now, we can see that we have a common factor of (3x - 2) in the numerator:

[-1 * (x^2 - 2x - 7)(3x - 2)] / [(x^2 - 4) * (x^2 + 2x)]

Now, we can factor the quadratic expression in the numerator by finding two numbers whose product is -7 and whose sum is -2. These numbers are -7 and 1.

So, the numerator becomes:

[-1 * (x + 1)(x - 7)(3x - 2)] / [(x^2 - 4) * (x^2 + 2x)]

Lastly, we can further simplify the denominator by factoring it as a difference of squares:

[(x + 1)(x - 1)(x - 2)(x + 2)] / [(x - 2)(x + 2)(x^2 + 2x)]

Canceling out the common factors:

[(x + 1)(x - 1)] / [(x^2 + 2x)]

Thus, we have obtained the simplified expression:

[(x + 1)(x - 1)] / [(x^2 + 2x)]