<s>[INST] Question about asymptotes. 
Let $f(x) = 3\cdot\frac{x^4+x^3+x^2+1}{x^2+x-2}.$ Give a polynomial
$g(x)$ so that $f(x) + g(x)$ has a horizontal asymptote of $0$ as $x$
approaches positive infinity.

I've tried using that if the degree of the denominator is bigger than the degree of the numerator, the horizontal asymptote is $y = 0$, but I couldn't get anywhere. How should I solve it?
 [/INST] Perform the Euclidean division of the numerator by the denominator:
$$x^4+x^3+x^2+1=(x^2+3)(x^2+x-2) -3x+7. $$
We deduce
$$f(x)=3(x^2+3)+3\frac{-3x+7}{x^2+x-2}.$$
The fraction tends to $0$ as $x\to\infty$, hence we can take $\:g(x)=-3(x^2+3)$.
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