Solution:

Total effort required by 4 painters for 37 days = 4 × 37 = 148 man-days.

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

Job still requires 148 − 8 = 140 man-days, and there are 20 painters available to do it.

Additional effort required by 20 painters for 140 man-days = 140 / 20 = 7 days.

Thus, 7 more days are required to complete the painting job.

Alternative Shorter Solution:

4 painters require 37 − 2 = 35 more days to complete the job.
Therefore, 4 + 16 = 20 painters will require 35 × 4 / 20 = 7 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?