When students first meet unique-factorization rings, usually @, @[i], and F[x], where F is a field, they may get the impression that non-unique-factorization rings are rare. After all, the text usually presents only one or two examples of such rings. Actually, with just a little more effort, we can offer an in£inite number of such rings. We describe two ways to do this. The first could be presented at the beginning of the discussion of unique factorization; the second depends on the fact that if R is a unique-factorization ring, then factorizations in R[x] and F[x], where F is the field of fractions of R, are the same up to constant factors. For our first source of examples let t E Z be an odd, negative integer, t < 3. Then @[4] is a non-unique-factorization ring. To show this, consider the product (1 + )(1 ) = 1 t, which is an even integer greater than or equal to 4. Let +(a + b4) = (a + b4)(a-b4) = a2 _ tb2, the usual norm. Using this norm, one shows that 2 has no non-trivial factorization in Z[4]. However, it is clear that 2 divides neither 1 + 4 nor 14 in Z[4], though it divides their product. Thus 1 t has at least two different factorizations as products of primes, so Z[4] is a non-unique-factorization ring. This method does not work when t is an odd positive integer, since, for example, @[4] and @[v] are unique-factorization rings. (They are some of the rings of algebraic integers that are Euclidean.) Our second source of examples depends on the fact that if R is an integral domain such that R[x] contains a polynomial whose factorization into primes in R[x] is essentially different from its factorization in F[x], then R is a non-uniquefactorization ring. Using this observation, we show that if t E Z is not a square and is of the form 4k + 1, k E , then R = @[4] is a llon-unique-factorization ring. Consider the polynomial Q(x) = x2 + x + (1 t)/4, which lies in R[x]. Let F be the field of fractions of R. Q(x) has in F[x] the factorization