To factor 10m^2n + 30mn^2 + 5mn by finding the greatest common factor (GCF), we need to identify the largest factor that all three terms have in common.
First, we can factor out a common factor of 5mn from all three terms:
10m^2n + 30mn^2 + 5mn = 5mn(2m^2 + 6mn + 1)
Therefore, the GCF of 10m^2n, 30mn^2, and 5mn is 5mn, and we can write the polynomial as the product of 5mn and the expression (2m^2 + 6mn + 1).