Polynomial multiplication , base x

For representing polynomials we can use base notation , it will make some operation easier.
for example 1+x+x^2 = (111)_x

For multiplication (x+1) * (x+1) = x^2 + 2x + 1
We can represent as (11)_x + (11)_x = (121)_x
It becomes like a regular multiplication if it does not excess 10 for one digit , but if it does than we need to seperate number e.g. (1,13,1)_x = x^2 + 13x + 1
We can assume at this point x > R (all real numbers) in base notation.