The simplest way to do these types of questions is to start with the Number at the front of the Monomial. In this case the numbers are:
20 and 4.
Factor these to primes:
20 = 2 x 2 x 5
4 = 2 x 2
Keep in mind that the LCM is the smallest number that you can divide both numbers (4 & 20) into with no remainder. It also happens to be the shared prime factors combined. It might in this case be obvious that the LCM is 20, but you can also see this from the fact that if you combine the prime factors, you'll also get 2 x 2 x 5.
The easy part is the exponents. You simply need the largest exponent from any of the combined monomials. When you think about this it should make sense because for example, x^3 = x * x * x. Any smaller exponent like x^2 is by definition going to be evenly divisible since division by an exponent with the same base is subtraction. Just find the largest exponent for each variable:
You have 20y^3 vs 4y^4. y^4 is larger than y^3.
You have 20^x^4 vs 4x^9. x^9 is larger than x^4.
So your final LCM is:
20y^4x^9