Here is how you can make sure there is a mistake in the book:
- Use the algorithm from the book to transform the regular expression into an NFA.
- Optional: determinise the automaton, using the algorithm from the book.
- Run your example through the automaton and note whether it accepts.
Provided you followed the conversion algorithm correctly and the word is not accepted, there is a mistake in the book: either the claim you quote is wrong or one of the algorithms is wrong.
Spoilers: as mentioned in the comments, the regular expression does not describe the claimed language. The author's epxression enforces "every $a$ is followed by a $c$", which is stronger than the claim.
Your expression looks quite complicated. All you need to do is ensure that every $b$ is followed by $a$, $b$ or the end of the word!
$(b^*a \mid c )^*b^*$
You can also build a suitable automaton -- basically, start with one that accepts everything and modify it to move to a dead state upon reading $bc$ -- and convert it to a regular expression. Perform whatever construction you can more easily prove correct.