Solution:

Total effort required by 4 painters for 34 days = 4 × 34 = 136 man-days.

Initial effort by 4 painters for 2 days = 4 × 2 = 8 man-days.

Job still requires 136 − 8 = 128 man-days, and there are 16 painters available to do it.

Additional effort required by 16 painters for 128 man-days = 128 / 16 = 8 days.

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

Alternative Shorter Solution:

4 painters require 34 − 2 = 32 more days to complete the job.
Therefore, 4 + 12 = 16 painters will require 32 × 4 / 16 = 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?