well, let's work backwards:

8 + 12z/(z+(1/2)) + -16/(z+(1/4)) =

[8(z+(1/2))(z+(1/4)) + 12z(z+(1/4)) + -16z(z+(1/2))]/((z+(1/2))(z+1/4)) =

[8(z^2 + (3/4)z + (1/8)) + 12z^2 + 3z - 16z^2 - 8z]/((z+(1/2))(z+1/4)) =

[8z^2 + 6z + 1 + 12z^2 + 3z - 16z^2 - 8z]/((z+(1/2))(z+1/4)) =

[(8+12-16)z^2 + (6+3-8)z + 1]/((z+(1/2))(z+1/4)) =

(4z^2 + z + 1)/((z+(1/2))(z+1/4)), and voila! there it is. so to get from this to your intended destination, read bottom-to-top.