That card seems to be supported by the "b44" driver.
and then check if there is now a "eth0" network device by running
If so, you can tell Slackware to use the b44 driver every time when the computer boots; run these commands to create the file "/etc/rc.d/rc.netdevice" where your network driver will be loaded:
echo "/sbin/modprobe b44" >> /etc/rc.d/rc.netdevice
chmod 755 /etc/rc.d/rc.netdevice
Next time the computer boots, it will load the driver. You still might need to configure your card - I do not know if you entered IP Address, netmask, etc, or indicated that you use DHCP to get an address. If you did that already during installation, then the above steps are sufficient - try them out by rebooting your computer and checking the network availability.