To solve this problem, we need to first perform the division and then the multiplication.

Step 1: Simplify the division
(3/4 + 5/6) / (1 1/2 - 3/5)
Convert the mixed numbers to improper fractions:
(3/4 + 5/6) / (3/2 - 3/5)
Find a common denominator for the fractions in the numerator and denominator:
(9/12 + 10/12) / (15/10 - 6/10)
Combine the fractions:
(19/12) / (9/10)
Multiply by the reciprocal of the denominator to divide:
(19/12) * (10/9) = 190/108 or 95/54

Step 2: Multiply by 1 8/19
Now, we will multiply the result from above by 1 8/19:
95/54 1 8/19
Convert the mixed number to an improper fraction:
95/54 (19 + 8)/19
95/54 27/19
Multiply the fractions:
(95 27) / (54 * 19) = 2565/1026 or 855/342

Step 3: Subtract the next expression
Now, we will subtract (1 - 7/8) 4:
Convert the mixed number to an improper fraction:
1 - 7/8 = 8/8 - 7/8 = 1/8
Multiply the fraction by 4:
1/8 4 = 4/8 = 1/2

Step 4: Subtract the results from step 2 and step 3:
855/342 - 1/2
Find a common denominator:
(855/342) * (171/171) - (171/342)
Now, subtract:
(855 - 171) / 342 = 684/342 = 2

Therefore, the final answer is 2.