Solution:

Total effort required by 4 painters for 10 days = 4 × 10 = 40 man-days.

Initial effort by 4 painters for 4 days = 4 × 4 = 16 man-days.

Job still requires 40 − 16 = 24 man-days, and there are 12 painters available to do it.

Additional effort required by 12 painters for 24 man-days = 24 / 12 = 2 days.

Thus, 2 more days are required to complete the painting job.

Alternative Shorter Solution:

4 painters require 10 − 4 = 6 more days to complete the job.
Therefore, 4 + 8 = 12 painters will require 6 × 4 / 12 = 2 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?