Basic Question on Finding Poles and Zeros
Write the equation in positive powers of z; factorise, if required, and simplify; identify the poles and zeroes by inspection.

You've already gone most of the way, the final step is:

\$X(z)=\dfrac{1}{(2z)^4(2z-1)}=\dfrac{1}{32z^4(z-0.5)}\$

Which gives four poles at \$z=0\$ and a pole at \$z=0.5\$.

There are no finite zeroes.

The way the algebra is 'done' is irrelevant, there is only one correct outcome! But some forms of an equation may make it easier to identify the features you're looking for.