Answer
$$1$$
Work Step by Step
We're given the following, in $\mathbb{Z}_3$:
$$2(2+1+2)$$
First, using the associativity of addition, group the terms:
$$2((2+1)+2)$$
Given that $2+1=0$ in $\mathbb{Z}_3$, we can simplify it to:
$$2(0+2)$$
Using the identity property of addition, this simplifies to:
$$2(2)$$
And computing the product, we get:
$$1$$
