You can put this solution on YOUR website!
I have found that the following can help one learn how to find LCM's (and GCF's for that matter).
First factor each expression fully (including factoring numbers into prime factors).
Then write the factors as you see below, rearranging the order as needed, so that the common factors, if any, lined up in columns:
2x^2y = 2 * x * x * y
3x^2+12 = x * 3 * (x+4)
Each column represents part of the LCM. So:
LCM = 2 * x * x * y * 3 * (x+4)
Note how the column with two x's, only contributes one x to the LCM.
Not only is this a visual way to find LCM's (and GCF's) but it can be useful in additional ways. For example, if you were finding the LCM of these expressions because they were denominators and you wanted to add or subtract the fractions, then not only is the LCM the lowest common denominator (LCD) but you can literally see what each denominator needs to turn it into the LCD. Comparing the factored form of 2x^2y to the factored form of the LCM you can see that it missing factors of 3 and (x+4). So you would multiply the numerator and denominator of that fraction by 3*(x+4). Similarly you can see that the 3x^2+12 denominator is missing factors of 2, x and y. So that fraction's numerator and denominator should be multiplied by 2xy.
BTW: The GCF is the product of all the factors that are in both sets of factors. In this example, only the second column is "full". So the GCF for your two expressions is just "x". (Again, each column contributes only one factor to the LCM or GCF.)