To simplify the expression 213\(_4\) x 23\(_4\), we first need to convert the numbers from base 4 to base 10. We can do this by multiplying each digit by the corresponding power of 4 and adding the results. For example, 213\(_4\) can be converted to base 10 as follows:

2 x 4^2 + 1 x 4^1 + 3 x 4^0 = 32 + 4 + 3 = 39

Similarly, we can convert 23\(_4\) to base 10 as follows:

2 x 4^1 + 3 x 4^0 = 8 + 3 = 11

Now, we can multiply the two numbers in base 10:

39 x 11 = 429

Finally, we need to convert the result back to base 4. We can do this by repeatedly dividing the number by 4 and writing down the remainders until the quotient is zero. The remainders, read in reverse order, give us the result in base 4:

429 ÷ 4 = 107 remainder 1

107 ÷ 4 = 26 remainder 3

26 ÷ 4 = 6 remainder 2

6 ÷ 4 = 1 remainder 2

1 ÷ 4 = 0 remainder 1

So, the result of multiplying 213\(_4\) x 23\(_4\) is 12231