How do you simplify #(x^2-x-12)/(x^2-11x+30)-(x-4)/(18-x)#?

I'm not sure how you are wanting this to simplify, but this should get you going. If this expression was equal to zero, you could add the second fraction to both sides and then divide by common factors (such as (x-4)).

To simplify the expression (x^2-x-12)/(x^2-11x+30)-(x-4)/(18-x), we first factor the denominators and combine the fractions with a common denominator.

The denominator x^2-11x+30 can be factored as (x-6)(x-5), and the denominator 18-x can be rewritten as -(x-18).

Now, we can rewrite the expression as follows:

(x^2-x-12)/[(x-6)(x-5)] - (x-4)/[-(x-18)]

To combine the fractions, we need to find a common denominator, which is (x-6)(x-5)(x-18).

Multiplying the first fraction by (x-18)/(x-18) and the second fraction by (x-5)/(x-5), we get:

[(x^2-x-12)(x-18)]/[(x-6)(x-5)(x-18)] - [(x-4)(x-5)]/[(x-6)(x-5)(x-18)]

Expanding and simplifying the numerators, we have:

[(x^3-19x^2+90x+12x^2-216x+1296) - (x^2-9x+20)]/[(x-6)(x-5)(x-18)]

Combining like terms in the numerator, we get:

(x^3-7x^2-125x+1276)/[(x-6)(x-5)(x-18)]

Therefore, the simplified expression is (x^3-7x^2-125x+1276)/[(x-6)(x-5)(x-18)].