I need help finding the greatest common factor between these two expressions 3u^3x^5y^8 and 21u^6x^5

I was thinking 7u^3y^8

Does 7 divide 3? Nope, so that cannot be a factor of 3u^3...

List all the factors of each term:

3 u^3 x^5 y^8
3 7 u^6 x^5

Now pick the highest power of each prime or variable that appears in both terms:

3 u^3 x^5

That is the GCF.