Solution:

Total effort required by 3 painters for 42 days = 3 × 42 = 126 man-days.

Initial effort by 3 painters for 2 days = 3 × 2 = 6 man-days.

Job still requires 126 − 6 = 120 man-days, and there are 15 painters available to do it.

Additional effort required by 15 painters for 120 man-days = 120 / 15 = 8 days.

Thus, 8 more days are required to complete the painting job.

Alternative Shorter Solution:

3 painters require 42 − 2 = 40 more days to complete the job.
Therefore, 3 + 12 = 15 painters will require 40 × 3 / 15 = 8 more days to complete the job.

Food for thought:

The number of days for the job is inversely proportional to the number of painters. So, is 'number of days × number of painters = constant'?

What assumptions were made to arrive at the solution?
Is something assumed about the output of the painters?