Just by inspection you can see that the LCM is 21. Why? Because 21 is a multiple of 7. This one's too easy to demonstrate the method on.
Find the prime factorization of both numbers:
15 = 3*5
50 = 2*5^2
Now, the LCM will contain ALL the prime factors of each number, only as many times as it shows up in the prime factorization.
So the LCM contains one factor of 2, one factor of 3, and two factors of 5. Thus:
LCM = 2*3*5^2 = 150.
We can't find prime factorizations of x and y since we don't know these numbers. So for the purposes of finding an LCM we treat them as if they were prime.
28xy^2 = 2^2*7*x*y^2
42x^2y = 2*3*7*x^2*y
LCM = 2^2*3*7*x^2*y^2 = 84x^2y^2