Question

How do you multiply `(a^2-a-12)/(a^2-5a+4)*(a^2+2a-3)/(a^2+a-6)`?

Answer 1

You start by first factorising the quatratic polynomials

`((a+3)(a-4))/((a-1)(a-4))*((a-1)(a+3))/((a+3)(a-2))`

`((a+3)(a-4)(a-1)(a+3))/((a-1)(a-4)(a+3)(a-2))`

After cancelling out all the like terms above and below the bar, we are left with:

`(a+3)/(a-2)` which cannot be further reduced.

Answer 2

You could just multiply the numerator and denominator polynomials separately (this would answer the question and if this is what is required please re-post)
but
it is easier if you factor your polynomials and simplify:
`(a^2-a-12)/(a^2-5a+4)*(a^2+2a-3)/(a^2+a-6)`

`= ((a-4)(a+3))/((a-4)(a-1))*((a+2)(a-1))/((a+3)(a-2))`

`= (color(red)(cancel((a-4)))color(blue)(cancel((a+3))))/(color(red)(cancel((a-4)))color(green)(cancel((a-1))))*((a+2)color(green)(cancel((a-1))))/(color(blue)(cancel((a+3)))(a-2))`

`= (a+2)/(a-2)`