The table represents the start of the division - 2x^3+11x^2-23x+20 of by x^2-3x+4. Which terms belong in one of the shaded cells in the table? Check all that apply.

SH = shaded

-2x 5

x^2 SH SH

-3x SH SH

4 SH SH

5

23x

5x^2

6x^2

2x^3

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.