When you multiply out, you are going to have a polynomial on top, and a polynomial below. First step is to figure out the degree of those polynomials. You don't have to worry about the exact coefficients. Just find the degree.
If the degree of the top polynomial is greater than that of the bottom polynomial, the limit will be $\infty$ or $-\infty$. If the degree of the top polynomial is less than that of the bottom polynomial, the limit is 0.
If the degree is the same, say $n$, then divide both the top and bottom by $x^n$. That reduces both the top and the bottom to the form $a+b/x+c/x^2+...$, and each of those is easy to evaluate as $x$ goes to $\infty$.
Now once you see that it is going to be in that form, you can notice that you don't have to bother multiplying it out to actually get the polynomials. You should be able to figure out what the degree will be without actually multiplying them out completely. Once you have the degree, say $n$, you can do the division of numerator and demoninator by $x^n$ as they stand, just distributing the $n$ $x's$ among the factors according to their degrees.
I don't want to fully solve your example since it is homework, but just to get you started, the degree is 4. The two factors in the top each get divided by $x^2$ as do each of the two factors in the bottom. The $(x^2+x)$ factor would then become $(1+1/x)$.
When you do that for all the factors, each factor will end up being something that goes to a constant as $x$ goes to $\infty$, and you should be able to see the limit.