How would we know the gcf of 4xy+8x^2y-24x^4y^5

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  4xy

Step-by-step explanation:

List the prime factors of each term.

Identify a factor that appears in each list. Cross it off of each list and add it to your list of "output" factors. Repeat until you run out of factors that appear in all of the lists.

You should quickly determine that all of the factors of 4xy appear in both other lists. That is the GCF. When you have exhausted the factors of 4xy, there are no more possible common factors.

  GCF = 4xy

Then the expression can be factored as ...

  = 4xy·(1 +2x -6x^3y^4)

Additional comment

If you use exponents to simplify writing repeated multiplication, you will find that the GCF has the factors with the lowest exponent. Here, the lowest power of 2 is 2^2 in 4xy; the lowest exponents of x and y are 1, also in that term.