How to know my wireless card has injection enabled?

From the output you displayed above, your current driver is not capable of packet injection. This is because you probably used the default (closed-source) drivers, many of which do not support injection.

You need to use the compat-wireless package, to compile your own drivers and use those instead (after uninstalling your current drivers). See this page from the Aircrack-ng Wiki which details how to compile your own drivers, as well as patch them to allow packet injection. At minimum, you need the mac80211.compat08082009.wl_frag+ack_v1.patch patch, as detailed in the wiki article linked above. Note that you should call the driver select script before compiling/installing.

As a side note, these drivers will also change the interface name from eth1 to wlan0 (which is also a lot more intuitive name). Depending on your card, the monitor interface name might be mon0 (increasing in number with each additional monitoring interface you create with airmon-ng start).

airmon-ng start eth1

This will create a new interface: mon0. You can then start your tools pointing to the new interface; i.e.

airodump-ng start mon0

Though the first poster is correct in saying you have wrong drivers – find the open source drivers that will change your driver name to wlan0 as the aircrack suite is a wireless pen-testing tool environment. There are other tools for LAN testing.

I was having no injection results. The solution was to invoke airmon as the following: airmon-ng start wlan0 <channel>. That will lock the mon0 interface to the channel <channel>. After that, I did a test and I got the result: Injection is working!

That is documented in the wiki:

Make sure the card(s) are on the same channel as your AP and locked on
this channel. When putting your card into monitor mode, be sure to specify
the channel via airmon-ng. You can use iwconfig to confirm which channel your
card is currently on. The injection test will fail if your card and access point
are on different channels.

Make sure your card is not channel hopping. A very common mistake is to have
airodump-ng running in channel hopping mode. If you use airodump-ng, be sure 
to use the ”-c <channel>” option. Additionally, ensure all network managers  
and similar are killed off.

..it would appear that your card is in "managed mode".In order to perform the test you will first have to put your card in "monitor" mode ...just search the internet.hope this help.