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That configuration *might* work, if
A. the jack used on the HP is the 'Line' or 'L' jack, not the 'P' or Phone jack. This is important.
B. there's a problem: the Vonage is not necessarily capable of supporting a fax connection. The 'digital' nature of the Vonage phone connection is not necessarily going to work with the HP fax. Analog phone lines are much more certain and likely to work.
Many digital phone services do not work with digital phone service of this type. But sometimes, it also helps to
C. set the HP device's fax settings to Low Speed, and to ECM Off (or, Error Correction Mode set to Off.) If at first the faxing does not work perfectly, or at all, try this settings change on the HP device's fax settings.