Where exactly are you stuck?

For starters, you're expected to look up the required transforms in a table. Letting LT[y] = Y, can you use your tables to get:

1. LT[y''] and LT[y']?

2. LT[cos(4t)]?

Substitute into the DE and make Y the subject, that is, get Y as a function of s. Can you do this?

When you've done all this things, please show what you've got (all working, please) and where you're stuck. If you can't do these things, please show what you can do and state where you're stuck.