I need help in this regural expression exercises
In order to solve this exercise, we need to come up for regular expressions for two classes of words:


Words of the form $x00$, where $x \neq \epsilon$ and the only occurrence of $00$ is at the end.
Words of the form $x01$, where $x \neq \epsilon$ and the only occurrence of $01$ is at the end.


Let us start with the first class. For starters, $x$ must end with $1$ or $2$. So such words are of one of the forms $y100$ or $y200$, where $y$ doesn't contain $00$. Presumably you already know how to describe all words not containing $00$ using a regular expression.

For the second class, the only constraints on $x$ are that it is non-empty and doesn't contain $01$. If you don't like the "non-empty" constraint, you can use the following case distinction:


If $x = y0$ or $x = y2$, then the only constraint on $y$ is that it doesn't contain $01$.
If $x = y1$, then $y$ cannot end with $0$. It could be empty. If it ends with $2$, then $x$ is of the form $z21$. If it ends with $1$, then we're again in the same situation: the preceding symbol (if any) cannot be $0$. Continuing in this way, we see that either $x \in 1^+$ or $x$ is of the form $y2z$, where $y$ doesn't contain $01$ and $z = 1^n$ for some $n \geq 1$.


I'll let you figure out how to describe strings $y$ avoiding $01$ – this is very similar to the case of avoiding $00$.