A lot of this website's regulars looked at that question and thought "Come on, that's way too easy." As you learn more, you will look back on this and then it will look a lot easier to you then than it does now.
At the same time you will also run into expressions for which it's not as obvious that they don't generate primes. I will use $f(n) = 4n^2 - 12n + 8$ to show you a couple of things you can do to find your way with the more difficult expressions.
Notice that there is only one variable, $n$, which is likely an integer, presumably positive. Still, I would try computing a few values, $-5 < n < 5$. This gives me: 168, 120, 80, 48, 24, 8, 0, 0, 8, 24, 48. It looks like $f(n)$ is never negative, even when $n$ is, and specifically, if $n < 0$ then $f(n) = f(|n| + 3)$.
Already this is enough to suggest that all $f(n)$ are divisible by 8. Try $$\frac{4n^2 - 12n + 8}{8} = \frac{n^2}{2} - \frac{3n}{2} + 1.$$ Not quite sure what to make of that, so let's try dividing by 4 instead: $$\frac{4n^2 - 12n + 8}{4} = n^2 - 3n + 2.$$
If you compute $n^2 - 3n + 2$ for a few small values of $n$, you will see that this expression also gives even numbers, though only some of them are also divisible by 4: 0, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, etc.
That sequence of integers looks familiar. Go to Sloane's OEIS and put it in: you won't get any results. Try dropping the first two zeroes and you'll get http://oeis.org/A002378 as the first result. These are the "oblong" numbers, $n^2 + n$, which are twice the triangular numbers.
In fact, if you search for $4n^2 - 12n + 8$ using the range $3 < n < 10$, you should get http://oeis.org/A033996 as the first result. That's the triangular numbers times 8.