Algebra Mess, don't know how to proceed

First, multiply (and divide) by $8(1+z^{-1})^3$: you get $$ \frac{8(1+z^{-1})^3}{8(1+z^{-1})^3+8(1+z^{-1})^2(1-z^{-1})+4(1+z^{-1})(1-z^{-1})^2+(1-z^{-1})^3}. $$ Next, expand the numerator to get $$ \frac{8(1+3z^{-1}+3z^{-2}+z^{-3})}{8(1+z^{-1})^3+8(1+z^{-1})^2(1-z^{-1})+4(1+z^{-1})(1-z^{-1})^2+(1-z^{-1})^3}. $$ Now, to expand the denominator. Let's do each term: $$ 8(1+z^{-1})^3=8+24z^{-1}+24z^{-2}+8z^{-3}; $$ $$ 8(1+z^{-1})^2(1-z^{-1})=8+8z^{-1}-8z^{-2}-8z^{-3}; $$ $$ 4(1+z^{-1})(1-z^{-1})^2=4-4z^{-1}-4z^{-2}+4z^{-3}; $$ and $$ (1-z^{-1})^3=1-3z^{-1}+3z^{-2}-z^{-3}. $$ Collecting terms, the denominator becomes $$ (8+8+4+1)+(24+8-4-3)z^{-1}+(24-8-4+3)z^{-2}+(8-8+4-1)z^{-3} =21+25z^{-1}+15z^{-2}+3z^{-3}. $$