Solution: By polynomial grid division, we start by the divisor 3x + 10 placed on the column headings. 3x 10 x^3 3x^4
We know that 3x^4 must be in the top left which means that the first row entry must be x^3. So the row and column multiply to 3x^4. We use this to fill in all of the first row, multiplying x^3 by the terms of the column entries. 3x 10 x^3 3x^4 10x^3 4x^2
We now got 10x^3 though we want 22x^3. The next cubic entry must then be 12x^3 so that the overall sum is 22x^3. 3x 10 x^3 3x^4 10x^3 4x^2 12x^3
Now we have 40x^2, so the next quadratic entry must be -3x^2 so that the overall sum is 37x^2. 3x 10 x^3 3x^4 10x^3 4x^2 12x^3 40x^2 -x -3x^2 -10x
This time we have -10x, so the next linear entry must be 3x so that the overall sum is 7x. 3x 10 x^3 3x^4 10x^3 4x^2 12x^3 40x^2 -x -3x^2 -10x 1 3x 10
The bottom and final term is 10, which is our desired answer. Therefore, we can now read the quotient off the first column: 3x^4+22x^3+37x^2-7x+10 / 3x + 10 = x^3 + 4x^2 -x + 1
