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the error was in the actual lesson walk through.
so it was teaching me the wrong way.

yes, i know about synthetic division.
i was almost certain i was correct, but wanted to make sure, that i was not making a mistake somewhere, with forgeting some obscure special case rule.
this...

I think I found an error in my math book.
I'm using polynomial long division to calculate the slant asymptote. can someone check this to see if i'm correct.

the book says the following.
...