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.

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