I don't think this is a good problem for metacognition. Solving it is too contingent on what people have taught you about irreducibility.
Anyway, as for your general question, I am sure you can find a trivial example over Z/mZ for some composite m. Also, I can't tell whether you want a solution to the specific question, so here it is: suppose that the given polynomial is equal to f(x) g(x), where f, g have integer coefficients. Then f(ak) g(ak) = 1 for all k. Since f(x) g(x) has no real roots, neither does f or g, so they cannot take both the values +1 and -1. Therefore suppose WLOG that f(ak) = g(ak) = 1 for all k. On the other hand, at least one of f, g has degree at most n. The problem is straightforward from here.
A bit of background: a few years ago, I designed such a course, after noticing that many of our social science majors were ending up taking a precalculus course (spending much time learning trig), which was mostly useless for their later study. I created a case-study approach to probability and statistics for students with weaker mathematics backgrounds (i.e., most of my students have been terrified by math in the past).
I wonder how much time the OP has to prepare -- it took me a long time (and a bit of grant support), as a pure mathematician to not only learn the basic probability, but more importantly to change my mindset from pure to applied mathematics. The formulas in basic probability and statistics are nearly trivial for a professional mathematician. The real work is identifying sources of statistical bias, interpreting results correctly, and accepting the fact that no studies are perfect. In statistical mechanics, you have an absurdly large sample size of molecules behaving in a very well-controlled environment. In practical statistics, you have a smaller, usually "dirtier" sample; as a pure mathematician, it's hard to accept this sometimes.
I'd begin preparing yourself by looking at three books -- the classic "How to Lie with Statistics", the new classic "Freakonomics", and Edward Tufte's "Visual display of quantitative information" (and/or his other books). None of these is directly relevant to your course material, but they will give you many ideas for teaching, for caution in the application of statistics, and for good and bad aspects of visual display of statistical information.
Directly regarding your questions: I'm not familiar with textbooks enough to advise you on this one (I wrote my own notes). But I strongly disagree with your assumption that "less diseases and more gambling" will make your class more engaging. Most people don't care about gambling; this is supposed to be a useful class, not training for a poker team. Real statistical studies are extremely interesting, especially given their life-and-death importance. Your students should be able to answer questions like "what is the probability a person has HIV, if their test result is "reactive"? How does the answer differ for populations in the U.S. vs. Mexico vs. South Africa?". Diseases, discrimination, forensic testing, climate extremes, etc.., are important issues to consider.
You might find gambling more interesting than diseases, but a teacher of math for social science students has a responsibility to approach important questions, and not contrived examples. Take it seriously!
There are many activities that you can enjoy with students. You might play a version of the Monty Hall game, for one. There are many activities with coin-tosses (e.g., illustrating the central limit theorem). You can illustrate sample size effects, by randomly sampling students in the course. You can certainly find cases in the media and recent studies, and use them as jumping-off points for discussion: you can even find funny ones in magazines like Cosmo, or on CNN.com so that the students can practice picking apart statistical arguments and deceptive rhetoric. I often make students find an article in popular media (like NYTimes.com) that refers to a study, then track down the original study, and compare the media summary to the published study to analyze how statistics are used and misused. This can make great classroom discussion.
Finally, it might be personal taste, but I would place a heavy emphasis on probability, especially Bayesian probability. Otherwise, the course can become mechanical and reinforce a common malady: students will think of statistics as the process of collecting data, putting data through a set of software/formulas to compute correlation coefficients, p-values, standard deviations, etc.., interpreting these numbers as facts about nature, and being done. The Bayesian approach, I think, requires more thought in setting up a problem, and yields more applicable results. In particular, there are significant Bayesian criticisms of "null-hypothesis statistical testing" that is the centerpiece of many studies; especially the overreliance on p-values is disturbing to me, and you might want to include criticisms of such things.
Best Answer
I gave a couple of lectures to some bright high-school students on the probabilistic method. For the lectures, I created a handout of 38 problems with hints at the end. The link is here.
I wouldn't say the problems are easy (many come from olympiads), but maybe some will suit your purposes, especially if you provide the hints.