Can you mix speedlights and studio lights in one lighting setup?

Yes. You can mix portable flashes/speedlights and studio strobes in the same setup, and trigger them with radio triggers, sync cords, or optical slaves.

A few things to watch for:

• TTL/preflash can complicate optical triggering, because some lights may fire on the preflash. Manual flash is usually simpler, or use optical triggers that ignore preflash.
• If you mean mixing flash with continuous studio lights, it also works, but flash is usually so much brighter during the exposure that the continuous light may contribute very little unless it is very powerful or the flash is kept relatively weak.
• Different light types may have different color temperatures, so you may need gels or white-balance correction.

In practice, you can use whatever combination gives the result you want. The key is deciding which light is your main source and balancing the others around it. For still life, continuous light can be fine; for people, flash is often preferred because of its power and ability to freeze motion.