What is the quotient in simplest form state any restrictions on the variable z^2-4/z-3/z+2/z^2+z-12?

The expression is:

       (z^2 - 4)
      
      
          z - 3
 

          z + 2
    

       z^2 + z - 12

You can factor the numerator z^2 - 4 = (z + 2) (z - 2)

And the denominatior z^2 + z + 12 = (z + 4)(z - 3)

That permits to write the quotient as:

        (z + 2)(z - 2)
      
      
             z - 3
 

             z + 2
    

       (z + 4)(z - 3)

Now you can multiplicate the numerator of the numerator times the denominator of the denominator, and multtiplicate the denominator of the numerator times the numerator of the denominator to obtain:

(z + 2)(z - 2)(z + 4)(z - 3)


       (z - 3)((z + 2)

Cancel the factors (z - 3) and (z + 2) because they are in both the numerator and the denominator =>

(z - 2)(z + 4) = z^2 + 2z - 8

The restrictions are that none of the cancelled factors can be 0, so z ≠ 3 and z ≠- 4.