To solve this problem, we need to first convert all mixed numbers into improper fractions.

5/12 + 2 3/7 can be written as:

5/12 + 2 (7*2+3) / 7
5/12 + 2 (17) / 7
5/12 + 34/7

Next, we need to find a common denominator between 12 and 7, which is 84.

(57 + 1234) / 84
(35 + 408) / 84
443 / 84

Now, let's simplify 5/12 * 3 4/5:

5/12 3 (53+4) / 5
5/12 3 19 / 5
(53 + 12*19) / 60
(15 + 228) / 60
243 / 60

Now, let's add the two results together:

443 / 84 + 243 / 60

To add these fractions, we need to find a common denominator between 84 and 60, which is 420.

(4435 + 84243) / 420
(2215 + 20412) / 420
22627 / 420

Now, we can simplify the fraction:

22627 / 420 = 4633 / 84

Therefore, the final result is 4633 / 84.