multiply and simplify (x^2+x-12/x-3)*(x^2-2x-3/x^2+5x+4)

((x^{2} + x - 12)/(x - 3))·((x^{2} - 2x - 3)/(x^{2} + 5x + 4))

First, if possible, factor the polynomial(s) in each rational expression:

(x^{2} + x - 12)/(x - 3) = ((x + 4)**(x - 3)**)/**(x - 3)**

Notice that in this rational expression there is a common factor in both the numerator and the denominator, that being 'x - 3'. With that, these 2 factors cancel each other out.

((x + 4)**(x - 3)**)/**(x - 3)** = (x + 4)

For the second rational expression,

(x^{2} - 2x - 3)/(x^{2} + 5x + 4) = (**(x + 1)**(x - 3))/(**(x + 1)**(x + 4))

Common factor in both numerator and denominator here is 'x + 1'

(**(x + 1)**(x - 3))/(**(x + 1)**(x + 4)) = (x - 3)/(x + 4)

Substitute the simplified forms of the rational expressions into the original problem:

((x^{2} + x - 12)/(x - 3))·((x^{2} - 2x - 3)/(x^{2} + 5x + 4))

= (x + 4)·((x - 3)/(x + 4))

= (**(x + 4)**(x - 3))/**(x + 4)**

After multiplying the simplified forms of the rational expressions, we see that there is another common factor in both the numerator and the denominator that cancels out, that being 'x + 4'. And that leaves us with the final solution:

(**(x + 4)**(x - 3))/**(x + 4)** = (x - 3)