To solve this expression, we need to follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

First, let's simplify the given expression:

(-21/6 -31/4): 23/5 + 2/3 21/4 - 5 1/6
= (-21/6 - 31/4) ÷ 23/5 + 2/3 21/4 - 5 - 1/6
= (-7/2 - 31/4) ÷ 23/5 + 2/3 21/4 - 5 - 1/6
= ((-14/4 - 31/4) ÷ 23/5) + (2/3 21/4) - 5 - 1/6
= (-45/4 ÷ 23/5) + (42/12) - 5 - 1/6
= (-45/4 5/23) + 21/6 - 5 - 1/6
= (-225/92) + 21/6 - 5 - 1/6
= -225/92 + 42/12 - 5 - 1/6
= -225/92 + 42/12 - 60/12 - 2/12
= -225/92 + (42 - 60 - 2)/12
= -225/92 + (-20)/12
= -225/92 - 20/12
= -225/92 - 15/6
= -225/92 - 30/12
= -225/92 - 60/24
= (-225 24 - 60 92)/(92 24)
= (-5400 - 5520)/(2208)
= -10920/2208
= -2730/552

Therefore, the expression simplifies to -2730/552 or -15/23.