Update: I was wrong but I'm getting somewhere
Update: Ok! So as z=2-3i was a root, z=2+3i (the conjugate) as Edgemaster says, is also a root. Therefore, (z-2+3i) and (z-2-3i) must be factors. Multiply the brackets out and you get (z^2 -4z +13) is a factor. Next step I guess is algebraic long division!
Update: Alright, so algebraic long division worked! So proud of myself right now, ahaha (I'm not doing further maths) so I proved z^2 -6z+34 was a factor. I'm guessing you know how to do algebraic long division, so basically put z^2 -4z +13 on the 'outside' and f(z) on the inside. Do the division, and on the top you should get the factor! And doing the division makes it easy to see how to do part b.
Update: So I got Q= 214? Hopefully you will get the same!
Update: (Last one I promise!) I found the other roots of the equation to be z=3+5i and z=3-5i
... Sorry I didn't see your last post
@RuneFreeze: you don't need to prove the remainder is zero, as long as you have 442 as the constant I think it's OK. You just need to show that the factor will be on the top. It only asks you to find Q in part b, so I think you can get away with not having zero as the remainder in part a.