How do I calculate this limit of factorial?

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I am having trouble solving this problem. If anyone could help me, I would be so thankful.

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Good day, to me there's a simple and informal way to solve this problem, and please correct me if I go wrong.

We could factor n!, at least ideally, being n! the fastest function of n growing to infinity here. We can therefore ignore n^3 and n^4 and simply consider the ratio between the coefficients of this two fastest (and equal) functions: -2/3. That's the solution.

If you asked for a more detailed and rigorous solutions, and this is not for high school, then others on this site, more trustworthy than me, will give for sure an exact explanation on how to solve the question.

Hope to help, even a little

Bye! :)