Does the expression ((0 + 1) (0 + 1)*)* 00 (0 + 1)* denote the language L(r) = {w ∈ Σ*: w has at least one pair of consecutive zeros}, for Σ = {0, 1} ?

Does the expression ((0 + 1) (0 + 1)*)* 00 (0 + 1)* denote the language L(r) = {w ∈ Σ*: w has at least one pair of consecutive zeros}, for Σ = {0, 1} ? ( an introduction to formal language and automata- Peter Linz Exercise 2, section 3.1 )

Solution:

Yes, because in the expression ((0 + 1) (0 + 1)*)* 00 (0 + 1)*, there are  two consecutive 0's (in bold letter),

and also the terms (in bold letter)  ((0 + 1) (0 + 1)*)* 00 (0 + 1)* can be null, because * contains null string too.