Based on the brothers' conversation, what can you infer about Robert's relationship with his mother and father?

We can infer that Robert’s father was not supportive of his decision to attend college. This is evidence in the brother’s conversation:  

ANDREW. You should have gone back to college last fall, like I know you wanted to. You’re fitted for that sort of thing—just as I ain’t.

ROBERT. You know why I didn’t go back, Andy. Pa didn’t like the idea, even if he didn’t say so; and I know he wanted the money to use improving the farm. And besides, I’m not keen on being a student, just because you see me reading books all the time. What I want to do now is keep on moving so that I won’t take root in any one place.

ANDREW. Well, the trip you’re leaving on tomorrow will keep you moving all right. (At this mention of the trip they both fall silent. There is a pause. Finally ANDREW goes on, awkwardly, attempting to speak casually) Uncle says you’ll be gone three years.

ROBERT. About that, he figures.

ANDREW. (moodily) That’s a long time.

ROBERT. Not so long when you come to consider it. You know the Sunda sails around the Horn for Yokohama first, and that’s a long voyage on a sailing ship; and if we go to any of the other places Uncle mentions—India, or Australia, or South Africa, or South America—they’ll be long voyages, too.

ANDREW. You can have all those foreign parts for all of me. (After a pause) Ma’s going to miss you a lot, Rob.

ROBERT. Yes—and I’ll miss her.

ANDREW. And Pa ain’t feeling none too happy to have you go—though he’s been trying not to show it.

ROBERT. I can see how he feels.

We also learn that Robert plans to be gone for several years. It is clear that his mother and father will miss him and likely worry about him during that time.

Explanation:

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