In how many ways can 7 distinct red balls and 5 distinct blue balls be placed in a row such that all red balls are adjacent,

The blue can be placed before the red or the red before the blue.
7 distinct balls have a permutation of 5040 while 5 distinct balls have a permutation of 120.
Each permutation of the reds can be combined with each permutation of the blues giving 5040×120=604800.
Double this to 1209600 to allow for the two positioning for red and blue.