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:
-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.
