I built a room next to a bathroom exterior wall where the bathroom and dryer vents terminated (on the exterior side). With the new adjacent room, the vents would have to be extended by 10 feet. My question is, can I connect the bathroom and dryer vents together and then extend one 4" vent pipe to the new exterior wall? Thanks in advance!

In this particular case, no, one common exhaust duct cannot connect to more than one exhaust. Even thou the bathroom fan has a backflow damper, the dryer will force air into the exhaust fan. If the fan was to be used at same time, the force of the dryer CFM is much more powerful than the exhaust fan, and the bathroom fan could not overcome the force.

The dryer, at least most that I know, does not have a backflow damper, some of the bathroom exhausted air would be forced into the dryer.

Only systems rated for common exhaust, one with a exterior mounted extractor fan, that draws air rather the push air, can connect to multiple vents.