Find a polynomial with certain conditions.

Question

Suppose that: $$f(x) = 3\frac{x^4+x^3+x^2+1}{x^2+x-2}.$$ Find a polynomial $h(x)$ such that $f(x) + h(x)$ has horizontal asymptote of 0 as $x$ approaches positive infinity.

Answer 1

Hint: Using polynomial division, we get $f(x) = 3x^2+9+\dfrac{-9x+21}{x^2+x-2}$

The $\dfrac{-9x+21}{x^2+x-2}$ term tends to $0$ as $x$ approaches $\infty$.