Question 604493
I've looked at your work, and you have solved the problem. However, the way you have written down your work is confusing and because you solved the problem yet did not know where to go next, I'm not sure that you fully understand what you are doing. 
.
If that is the case, then let's try a different way of doing the problem. It may help you to understand what you are doing.
.
One way of doing long division problems such as this one is to work it very similar to the long division you learned when you were working with just numbers. Write out the divisor, then put a division sign and inside it put the dividend.When you do that for this problem you have:
.
""""""""""""________________________
-x^2 – x – 5 | 4x^3 + x^2 + 17x – 15
.
Ignore the quote signs ("). I am just using those as spacers so that I can get things to line up and look the way that I want them to look.
.
Next, take the first term of the divisor (the -x^2) and divide it into the first term of the dividend (the +4x^3). The answer to this division is -4x. Write it over the long division sign.
.
""""""""""""" -4x
""""""""""""________________________
-x^2 – x – 5 | 4x^3 + x^2 + 17x – 15
.
Now just as you do in long division with numbers, you multiply the -4x times each of the terms in the divisor and you line the three answers up with the corresponding terms in the dividend as shown below:
.
""""""""""""" -4x
""""""""""""________________________
-x^2 – x – 5 | 4x^3 + x^2 + 17x – 15
""""""""""""" 4x^3 +4x^2 + 20x
"""""""""""""  ----------------------
.
Notice that the corresponding terms line up vertically and form columns involving the same powers of x.  (In other problems the dividend in the division sign might be missing a term. If it is, then insert a zero times the missing power of x as a place holder). Now do the algebraic subtraction in the vertical columns to get the answer shown below. And bring down the next term from the dividend, the -15. The result of this subtraction should look like this:
.
""""""""""""" -4x
""""""""""""________________________
-x^2 – x – 5 | 4x^3 + x^2 + 17x – 15
""""""""""""" 4x^3 +4x^2 + 20x
"""""""""""""  ----------------------
""""""""""""""""" 0 - 3x^2 - 3x -15
.
The results of that subtraction look familiar. You had it in your work. Next take the first term of the divisor (-x^2) and divide it into the first term of the subtraction results (-3x^2). The answer to that division is +3. Write this above the division sign:
.
""""""""""""" -4x + 3
""""""""""""________________________
-x^2 – x – 5 | 4x^3 + x^2 + 17x – 15
""""""""""""" 4x^3 +4x^2 + 20x
"""""""""""""  ----------------------
"""""""""""""""" 0 - 3x^2 - 3x - 15
.
Multiply the +3 above the division sign times each of the terms in the divisor and write the products below the subtraction results. I will look like this:
.
""""""""""""" -4x + 3
""""""""""""________________________
-x^2 – x – 5 | 4x^3 + x^2 + 17x – 15
""""""""""""" 4x^3 +4x^2 + 20x
"""""""""""""  ----------------------
""""""""""""""""" 0 - 3x^2 - 3x -15
""""""""""""""""""" - 3x^2 - 3x -15 
""""""""""""""""" -------------------
.
Subtract vertically in columns and note that each of the subtractions results in a zero as shown:
.
""""""""""""" -4x + 3
""""""""""""________________________
-x^2 – x – 5 | 4x^3 + x^2 + 17x – 15
""""""""""""" 4x^3 +4x^2 + 20x
"""""""""""""  ----------------------
""""""""""""""""" 0 - 3x^2 -3x -15
""""""""""""""""""" - 3x^2 -3x -15 
""""""""""""""""" -------------------
"""""""""""""""""""""""""" 0 0 0
.
There is nothing left. You have used all the numbers in the dividend ... all the way down to the -15. And after the subtraction there is nothing left that the first term of the divisor will go into at least once. You can do no more division and there is no remainder. This tells you that -4x +3 is the answer to this problem. If you divide -x^2 - x -5 into 4x^3 + x^2 + 17x – 15 you get an answer of -4x + 3, and you had that correct in what you did. You showed this in the last step of your work in that you multiplied -4x + 3 times the divisor -x^2 - x - 5 and got the dividend of 4x^3 + x^2 + 17x – 15. This is just like the checking you do when you divide numbers ... you multiply the quotient times the divisor to ensure that you get the dividend that is inside the long division sign.
. 
Maybe this will give you a little better insight into the problem and a way to document your work so you communicate clearly what you are doing when you solve it.
.
Hope this helps you to see what you are doing because it is somewhat similar to what you are familiar with when you long divide numbers.
.