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Let $y=f(x)=\frac{x^{4}+4 x^{3}+2 x^{2}}{x^{3}+x}$. Find all the asymptotes of the function, it they exist.

Step 1 ：Find the zeros of the denominator: \(x^3 + x = x(x^2 + 1) = 0\)

Step 2 ：The denominator has one real zero at \(x = 0\)

Step 3 ：Find the limit of the function as x approaches infinity: \(\lim_{x \to \infty} \frac{x^{4}+4 x^{3}+2 x^{2}}{x^{3}+x}\)

Step 4 ：The limit as x approaches infinity is infinity, so there is no horizontal asymptote

Step 5 ：Find the limit of the function as x approaches 0: \(\lim_{x \to 0} \frac{x^{4}+4 x^{3}+2 x^{2}}{x^{3}+x}\)

Step 6 ：The limit as x approaches 0 is 0

Step 7 ：\(\boxed{\text{The function has a vertical asymptote at } x = 0 \text{, and there is no horizontal asymptote}}\)