Is it possible to patent a software product and simultaneously open-source it, or are both strategies mutually exclusive?

It is certainly possible to attempt to (or actually obtain) a patent on your technology, in addition to releasing the software under an open-source license.

Patent rights would give you great control over who uses
your software. In contrast, the basic idea behind distributing
software under an Open Source license is that anyone
should be able to view and use the “source code” of the
computer program and modify it for his own purposes.
You would have to make the source code available to others, and then anyone could modify it without obtaining permission from you.

A business decision to distribute software under an open source license would affect how you are able to use your patent rights, but such a decision should not affect your ability to apply for patent protection. Releasing software this way may give certain patent licenses to individuals who receive the software, and you would not be able to control who those individuals are.

Whether releasing the software in an open source format is a good strategy for you depends on what business model you are trying to achieve. Typically, a customer of open source technology enjoys the value of their ability to use the software under standard commercial terms and support. They are willing to pay for the legal protection, such as indemnification from intellectual property infringement, as well as the professional support and training that are typical of commercial software built on top of the innovation and independence that comes with open source.