By polynomial grid division, we start by the divisor x²-3x+4 placed on the row headings of the table and end with the quotient -2x + 5 on the column headings as given. We know that -2x³ must be in the top left which means that the first column entry is indeed -2x. So the row and column multiply to -2x³. We use this to fill in all of the first column, multiplying -2x by the terms of the row entries.
        -2x   5
   x²   -2x³
   -3x    6x²
   4    -8x
We now got 6x² though we want 11x². The next quadratic entry must then be 5x² so that the overall sum is 11x². Multiplying 5 by the terms of the row entries, we fill in all of the second column:
        -2x   5
   x²   -2x³   5x²
   -3x   6x²  -15x
   4    -8x   20
The bottom and final term is 20, which is our desired answer and we can read the quotient off the first row:Â
   -2x³ +11x² - 23x + 20 / x² - 3x + 4 = -2x + 5
We have calculated for all the terms that belong in table, therefore, the terms 5x² and 6x² belong in the shaded cells.
