How can I count this example? I am having a hard time figuring it out.
Rather than just supplying an answer, which won't particularly help any other questions that are similar, I'll supply a nice simple way to achieve what you want.
In 4/4 time, there's room for eight quavers (eighth notes - called such because eight of them just fill one bar); one crotchet (quarter note) is worth two counts, and because a dot after a note makes it half as long again, it's worth a count of three.
So - the quaver rest and the follwing quaver get one count each.
The crotchet rest gets a count of two.
The dotted crotchet gets a three count, leaving just one count (the number 8) for the remaining quaver. I used to find it helped to actually write the numbers out, over the dots. Not enough room? Then write it all out wider!