what is quotient of (6-x)/(x^2 + 2x-3) divided by (x^2-4x-12/x^2 +4x +3? State restrictions. answer is -(x+1)/(x-1)(x+2) with restrictions of -3, -2,1,6. Don't understand why -3 and 6 are restrictions.

Question

what is quotient of (6-x)/(x^2 + 2x-3) divided by (x^2-4x-12/x^2 +4x +3?  State restrictions.  answer is  -(x+1)/(x-1)(x+2) with restrictions of -3, -2,1,6.  Don't understand why -3 and 6 are restrictions.

phi · Answer

when you divide by a fraction, you change the problem to multiply by its inverse (flip the fraction)
the problem is 
$$ \frac{(6-x)}{x^2+2x-3} \cdot \frac{x^2+4x+3}{x^2-4x-12} $$

after factoring we get
$$ \frac{(6-x)}{(x+3)(x-1)} \cdot \frac{(x+3)(x+1)}{(x-6)(x+2)} $$
it looks like we can cancel some terms.
however, the simplified fraction does not match *exactly* this expression, because this expression is not defined at x= -3, 1, 6 or -2
to be complete, we make note of these numbers.