Complex fractions, why does this shorter method immediately get a wrong answer?

The original problem is (4 + (1/x))/(3+(2/x^2)). The answer is (4x^2 + x)/(3x^2 + 2).

BUT my instructor claimed
4 + (1/x))/(3+(2/x^2)) = [x] (4 + (1/x)) / [x^2] (3+(2/x^2)) = (4x + 1) / (3x^2 + 2).
Update: And why can't it be done like that? Is it possible to change it so that it works and is consistent across other problems while also being useful? I think I vaguely remember something about factoring out as yet another fraction.
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