Initially- R-4 & B-6 balls.
1) if red is drawn then probability = 4/10. Then that red + 2 more red balls are added to bag. So now R- 6 and B-6. Now again 2nd time it is red then probability= 6/12
2) If black is drawn then (From initial R-4 nad B-6) then probability of black = 6/10. Then that drawn black + 2 more black balls are added into bag. Now again 2nd time ball is drawn and it is red, its probability= 4/12 (As R-4 and B-8).
Therefore probability of red drawn = 4/10 * 6/12 + 6/10 * 4/12= 48/120= 2/5.