Can the N300 use a wired connection?

Can I use my wired connection to extend the wireless?
I use a chromecast on my living room tv, which requires a wireless connection. I have a home with wired connections throughout the house, but my wireless doesn't reach some of the further corners of the house. I've put the extender closer to these areas, but it gets a poor signal from the router. I have connected the extender to the port with an ethernet cable, but it seems it is unable to broadcast the signal. It states I can connect to it but it can't reach the internet. Is there a way to make this work?

When I plug the extender closer to the router without a wired connection, it works fine, but doesn't reach the area I need wireless provided. At this point, I'm considering purchasing a router with greater range, but I'd like to work with what I have. Is this possible?