The only time that a graph has a slant asymptote is only when the degree of the top is one bigger than the degree on the bottom. If the degree was bigger on top by something more than one, like 2 or 3, on the top than the bottom then it is neither a horizontal nor a slant. A slant asymptote only exists if the degree on the numerator is one bigger. To find the equation of the slant asymptote you use long division. When you solve using long division everything in the equation is the slant asymptote except for the remainder of the equation. Graphs can cross through slant asymptote only towards the middle of the graph.
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