Please help me understand how resonance structures work. I thought I'd worked it out and understood it but once I'd seen this question, I was confused again. I am mostly confused on how to start on these problems; I know that I need to move the negative charge on the Carbon on the extreme left to another bond or atom but I don't understand how to do this without violating the octet rule. Please help me out! (the question is attached in the picture)

yes it is important that how to start resonance:
Always remember it is the delocalization of electron density ...so starting is from electron rich site (means negative or Lp or pi bond density) to the less electron rich site (positive charge of Electron withdrawing group)
In the structure ...resonance will start from LP on the carbon(negatively charged carbon) and will move in the direction of carbonyl group ...LP will make the pi bond and pi bond will make the LP. Most stable form will be with  negative charge on more Electronegative
atom.
Please check attached image: