Determine all the ideals, prime ideals, and maximal ideals of $\mathbb{R}[x]/I$ where $I$ is the ideal generated by $(x^2+1)(x-2)^2$.
I am currently doing some reading on ideals (see http://www.math.niu.edu/~beachy/aaol/rings.html#ideals), and I must say I am somewhat confused. The author describes what ideals are, but does not provide examples as to how to determine these in situations like the one above? What specific steps am I supposed to take to answer questions of the type which is shown above?
Somewhere in your notes you should find a result that links ideals of a domain $R$ containing some ideal $I$, to those of $R/I$.
Then you are left with the task of determining the ideals that contain $I$. To do this recall that you are working in a PID where each ideal is thus generated by a single element and inclusion of ideals is linked to divisibility of generating elements.