z³ - iz² + 3iz + 3

Factor z² out of the first two terms

z²(z - i) + 3iz + 3

Now we'd like to be able to factor z - i out of the last two terms 3iz + 3,
It looks hopeless but we have a magic trick.  Since 1 = -i² we can multiply 
the 3 by -i² since that equals 1 and will not change the value of the
expression:

z²(z - i) + 3iz + 3(-i²)

z²(z - i) + 3iz - 3i²

Now we can factor +3i out of the last two terms:

z²(z - i) + 3i(z - i)

That did it, so now we can factor out (z - i)

(z - i)(z² + 3i)

Edwin