For the umpteenth time, please accept هٰذه and هٰذا (and لكن while we're at it) without the dagger alif. It is impossible to type on a standard Arabic computer keyboard and completely unnecessary and I can't get the answer correct without copying it and pasting it from the correction, which defeats the purpose.
As far as I know, in the standard Arabic grammar, "هَذَا" hadhaa, "هَذِهِ" hadhihi and "لَكِنَّ" lakinna (or "لَكِنْ" lakin) are without dagger alif. So, it should be accepted (ie. if we don't write the dagger alif). Furthermore, originally Arabic is without any Harakaat.
"ghurfatu" غرفة is muDaaf مضاف of "2al-ustadhi" (which is definite) while "2al-ustadhi" is 2ism 3alam laqab اسم علم لقب of "muHammad(in)" محمد. In other word, we can say "muHammad(in)" محمد is a badal بدل of "2al-ustadhi" الأستاذ.
So, "ghurfatu" غرفة becomes definite by "2al-ustadhi" الأستاذ. And, we don't need to put ال on "muHammad" محمد because it is a person name whereby it is already a definite noun. For example, if the room is definite by the name, the phrase is Muhammad's room غرفةُ محمدٍ. (I am so sorry for my limited English terms).
Including the definite article ال here, is an Arabic grammar thing, iDafah, i think (google it). Plus, when we say professor Mohammad's room in English, professor is like a title, so that implies a specific Mohammed and including "the" would sound stilted. Couldn't use the English grammar to explain why, though.