Well, the Asus M5A78L-M LX motherboard specification indicates that it has a PCIe x16 (PCI Express) slot available, see:
http://www.asus.com/Motherboards/M5A78LM_LX/#specifications
Then, checking the AMD Radeon website for HD 7770 specifications it states that it is compatible with PCI Express x 16. see:
http://www.amd.com/uk/products/desktop/graphics/7000/7770/Pages/radeon-7770.aspx#3
Based on that I would guess it should work, But, note the other requirements shown regarding power connections / rating. You would need to have a suitable 6-pin 75W connection available from a minimum of 500W power supply. Plus recommended minimum of 4GB RAM, which is ok since you have 8GB.
Hope that helps.