X^2-9 over x^2+4x-12 divided by x^2+2x-3 over x^2+5x-6

Question

What do you get when you simplify this

Answer 1

First factorise all the components:

x²-9=(x-3)(x+3); x²+4x-12=(x+6)(x-2); x²+2x-3=(x+3)(x-1); x²+5x-6=(x+6)(x-1).

{(x-3)(x+3)/[(x+6)(x-2)]}/{(x+3)(x-1)/[(x+6)(x-1)]}=

{(x-3)(x+3)/[(x+6)(x-2)]}/{(x+3)/(x+6)}.

The division can be replaced by multiplication and inversion of the denominator fraction:

{(x-3)(x+3)/[(x+6)(x-2)]}{(x+6)/(x+3)}=(x-3)(x+3)(x+6)/[(x+6)(x-2)(x+3)]=(x-3)/(x-2).