When you are dividing by "x + 1" you use "-1" for the synthetic division. The "-1" goes on the outside of the synthetic division symbol and the coefficients of the polynomial go on the inside (those are "2", "10", "-1", "11" and "39" so it would look like this:
Bring the "2" down and multiply it by the -1 (the -1 on the left) and place the result under the "10":
-1 | 2 10 -1 11 39
| -2
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2
Then add the 10 and -2 to get 8, Place the 8 below the line next to the "2" you brought down:
-1 | 2 10 -1 11 39
| -2
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2 8
Then multiply the 8 by the -1 on the left like before and put the result (-8) under the -1:
-1 | 2 10 -1 11 39
| -2 -8
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2 8
Add the -1 and -8 and place the result below the line next to the -8:
-1 | 2 10 -1 11 39
| -2 -8
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2 8 -9
Multiply the -9 by the -1 again and place the result (9) below the 11. Add to get 20 and place the 20 below the line again:
-1 | 2 10 -1 11 39
| -2 -8 9
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2 8 -9 20
Multiply the 20 by the -1 again and place the result (-20) under the 39, then add:
-1 | 2 10 -1 11 39
| -2 -8 9 -20
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2 8 -9 20 19
The quotient is the polynomial made up of the coefficients on the bottom row. To determine the variable power, use one less than the power you started with in the original polynomial. Since the original polynomial started with x4, the quotient will start with x3 then just work your way down with x2, x and the constant term, applying each coefficient you got to each of these making the quotient:
2x3 + 8x2 - 9x + 20. The remainder is the last coefficient so remainder 19.
