I know that my question seems subjective but also it is possible that there may be a general rule which I do not know currently. While I am studying abstract algebra, sometimes I can not conclude the corollary of the theorems that are stated. I revise what I do in the related chapter, then think about the theorem, try to make claims about what can be said after the theorem. First, I do the work I mentioned above. Second, I try to prove the corollary. If I have enough time, I spend days to give a proof.
An example:
I was studying the PID's from Dummit & Foote's Algebra -which I have studied multiple of times before- $3$ days ago and saw the following theorem and its corollary:
Theorem. Every nonzero prime ideal in a PID is a maximal ideal.
Corollary. If $R$ is any commutative ring such that the polynomial ring $R[x]$ is a PID, then $R$ is necessarily a field.
I did not look at the proof and still working on it.(Using "tools" like an ideal $I$ is maximal then $R/I$ is a field etc.)
I go back, study again. Think, think, think. Use different proof methods.
What is the thing that I am not aware of? What can be done in my situtation?
There is no general method for finding the "obvious" proofs, other than general experience. However, I can tell you what in this specific situation hints toward the solution outlined in the comments to the question above. There are a lot of interconnecting facts that together point toward this solution.
An ideal is maximal iff you get a field when you divide out by it. Thus the theorem says something about what quotient rings you can get (i.e. if you get an integral domain, and it is not the ring itself, then it must be a field). Also, because $R[x]$ is an integral domain, $(x)$ is prime, and $R\simeq R[x]/(x)$ (this quotient is the main way of getting information about $R$ if you know a lot about $R[x]$ in the first place; we know more about quotients than about subrings). All these facts come together with the theorem itself, and the corollary is the result.