what is the quotient in simpliest form state the restrictions on the variable
z^2-4/z-3 divided z+2/z^2+z-12
i got z^2+2x-8 but am not sure of the restrictions maybe 3,-2
You are almost correct on the restrictions. At z=-2,3, both fractions are 0/0. However,
z^2+z-12 = (z+4)(z-3), so you know right off that z ≠ -4,3, so the restrictions are
z ≠ -4,-2,3 because at all 3 of those values the original fractions are not defined.
Knowing that, you have
(z-2)(z+2)/(z-3) * (z+4)(z-3)/(z+2) = (z-2)(z+4)
so your final polynomial is correct.