Separate the monomial into its individual constituents. In the case of the monomial, 12x^3y^2, the individual factors are 12, x^3 and y^2. X^3 and y^2 are both variables, and 12 is a constant.
Break down the constant into its prime factors. These are factors that cannot be broken down into real integers any further. Twelve breaks down into its factors of 3 and 4. Three is a prime number, and its only other factor is 1. Four factors out to 2 and 2. Two is a prime number. Therefore, the prime factorization of 12 is 3 * 2 * 2.
The variables x^3 and y^2 have degrees of 3 and 2, respectively. This means that the variable x^3 factors out to x * x * x and the variable y^2 factors out to y * y.
Putting all the factors together gives the factorization of the given monomial. The monomial 12x^3y^2 factors out to 2 * 2 * 3 * x * x * x * y * y. If the constant is negative, place a -1 in the constant factorization.