Polynomial long division

[Jherek2]

Hi, I have this long division exercise and I am completely stumped. This is the last in a set of 26, the others I have done without a problem, but I can't see how to start this one. Where I am stuck is in trying to divide the lead term by , which doesn't go exactly, so I end up with a fraction and I don't know how to continue with that. I guess that there's a really simple thing I can't see that would make it easy, so a hint to get me started would be appreciated...
()/()

[stapel_eliz]

Hi, I have this long division exercise and I am completely stumped. This is the last in a set of 26, the others I have done without a problem, but I can't see how to start this one. Where I am stuck is in trying to divide the lead term by , which doesn't go exactly, so I end up with a fraction and I don't know how to continue with that. I guess that there's a really simple thing I can't see that would make it easy, so a hint to get me started would be appreciated...
()/()

You're going to be stuck with fractions.

What do you get when you divide 17x^4 by -5x^2? You get -(17/5)x^2, so that's what goes on top.

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a)
               -(17/5)x^2
               ----------------------------------------------
-5x^2 - 4x + 2 )  17  x^4 +    2  x^3 -    39  x^2 - 16x + 10
                 
               ----------------------------------------------
                           

Multiply this by all three terms of the divisor:

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b)
               -(17/5)x^2
               ----------------------------------------------
-5x^2 - 4x + 2 )  17  x^4 +    2  x^3 -    39  x^2 - 16x + 10
                  17  x^4 + (68/5)x^3 - (34/5) x^2
               ----------------------------------------------
                           

Then subtract:

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c)
               -(17/5)x^2
               ----------------------------------------------
-5x^2 - 4x + 2 )  17  x^4 +    2  x^3 -    39  x^2 - 16x + 10
                  17  x^4 + (68/5)x^3 - (34/5) x^2
               ----------------------------------------------
                           -(58/5)x^3 - (161/5)x^2

Then carry down the remaining terms:

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d)
               -(17/5)x^2
               ----------------------------------------------
-5x^2 - 4x + 2 )  17  x^4 +    2  x^3 -    39  x^2 - 16x + 10
                  17  x^4 + (68/5)x^3 - (34/5) x^2
               ----------------------------------------------
                            (78/5)x^3 - (161/5)x^2 - 16x + 10

Then do the same thing for the next step.

[Jherek2]

Ah, thank you for that! I suspected it may be something like that, but it seemed so complicated, I was sure I was overlooking something simple.
I will give it a go and see if I can get it right!
Well the question in the book was find the remainder, which they gave as 2x^4, mine is below and try as I might I can't get 2x^4. I even put the question into Wolfram Alpha and got the same answer as mine, so I can't explain the 2x^4?

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               -(17/5)x^2 + (58/25)x + (573/125)
               ----------------------------------------------
-5x^2 - 4x + 2 )  17  x^4 +    2  x^3 -    39  x^2 - 16x + 10
                  17  x^4 + (68/5)x^3 - (34/5) x^2
               ----------------------------------------------
                           - (58/5)x^3 - (161/5)x^2  - 16x + 10
                           - (58/5)x^3 - (232/25)x^2 + (116/25)x^2
                           -------------------------------------------------
                                         -(573/25)x^2 - (516/25)x   + 10
                                         -(573/25)x^2 - (2292/125)x + (1146/125)
                                         ---------------------------------------
                                                      - (288/125)x  + (104/125) remainder