Calculating the Resistance using Mesh-Current Method
The answer is definitely wrong. Here's a quick way to tell. If you have R=infinite, then you have 40V/300ohms = 0.133 amps if you ignore current provided by the 20V supply. That's the absolute minimum current that will flow through I. Any lowering of R from infinite will only increase the amount of current flowing through I. That means that their initial statement of I=80mA is impossible.

The only exception to that is if we do allow negative resistances as you have calculated. A negative resistance would be kind of like a voltage/current source. You're likely correct in your negative resistance calculation.