You need to test if `z^2 - 8/z` yields `0` , hence, you need to start by bringing the members of expression to a common denominator, such that:
`z^2 - 8/z = 0 => z^3 - 8 = 0 => z^3 - 2^3 = 0`
You need to convert the difference of cubes into the following special product, such that:
`z^3 - 2^3 = (z - 2)(z^2 + 2z + 4)`
Since the problem provides the information that `z^2 + 2z + 4 = 0` , hence, by zero product rule, yields that `(z - 2)(z^2 + 2z + 4) = 0` , thus `z^3 - 8 = 0` .
Hence, testing if the given identity is `z^2 - 8/z = 0` valid yields that the equivalent expression `z^3 - 8 = 0` is valid, thus, the statement `z^2 - 8/z = 0` holds, under the given conditions.
Given `z^2+2z+4=0` ,we wish to prove `z^2-8/z=0`
`=((z-2)xx0)/z` because `z^2+2z+4=0` ( given )