Algorithm for multiplication in finite field

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I am trying to understand multiplication in finite fields. I have understood the usual multiplication method for multiplying two polynomials in a finite field. But I am unable to understand the algorithm using arrays, a portion of which from the training notes is attached. Also I have not understood the underlied step in the attached photo. Shifting of cells for multiplication by x is clear....but I have not understood the process if this is not the case. Any help towards understanding this concept would be appreciated enter image description here