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sqdancefan
You don't even have to actually do the division to find the correct answer.

The leading term of q(x) is x³/x² = x, so only the 1st and 3rd selections are appropriate.

The remainder r(x)/b(x) will have the divisor as b(x), so only ...
.. the 1st selection is appropriate.
mathmate
This can be done using long division that we have learned with numbers, except that each digit in the numeric long division is replaced with a number that can exceed 10.

Thus we can write the division as where the right most digit represent a number, and each number to the left represent an increasing power of x.
                 ________________________
1 +2 +1    |  1   +10   +13     + 39

We will use a first quotient digit of 1 and begin the division

                 _1______________________
1 +2 +1    |  1   +10   +13     + 39
                   1     +2     +1                                   (subtract)

                            8     12      39


After that, we have a multiplier of 8 and continue

                 _1___8___________________
1 +2 +1    |  1   +10   +13     + 39
                   1     +2     +1                                   (subtract)

                            8     12      39
                            8     16        8                         (subtract)

                                    -4      31

This means that we have a quotient of (x+8) with a remainder of -4x+31
thus the final expression looks like

[tex]\frac{x^3+10x^2+13x+39}{x^2+2x+1}=x+8+\frac{-4x+31}{x^2+2x+1}[/tex]

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