Identify the factors of each term.
8, x^2, y
12, x, y^2
20, x^2, y^2
All of the constants are divisible by 4, but by no larger number.
All of the x-factors are divisible by x, but no higher power.
All of the y-factors are divisible by y, but no higher power.
The greatest monomial factor is the product of these maximum divisors:
4xy
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= (4xy)(2x -3y +5xy)
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When you factor to primes, you are looking for the smallest power of each factor.
8 = 2³
12 = 2²·3
20 = 2²·5
2 is a common factor, and its smallest power is 2. So, 2²=4 will be in your monomial. 3 and 5 are not factors in all terms.
Likewise, x and x² are common factors, and the smallest power of x is 1. So, x will be in your monomial. Same deal for y.