<PRE><B>Question 474995</B>
&lt;pre&gt;  4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5&lt;/pre&gt;We will call the first expression &quot;dividend&quot; and the second &quot;divisor&quot;.
First thing to do is to look at the first term of both expressions,
4x&lt;sup&gt;3&lt;/sup&gt; and -x&lt;sup&gt;2&lt;/sup&gt;, and answer the question 
&quot;How much is (4x&lt;sup&gt;3&lt;/sup&gt;)/(-x&lt;sup&gt;2&lt;/sup&gt;)?&quot;
This answer, which is -4x, is the first term of the result. Hence, 1st step is:&lt;pre&gt;  4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5 = -4x ...&lt;/pre&gt;Next thing to do is to multiply divisor by this term just obtained (-4x) and write the result immediately below the dividend aligning the like terms below each other.&lt;pre&gt;  4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5 = -4x ...
  4x&lt;sup&gt;3&lt;/sup&gt; + 4x&lt;sup&gt;2&lt;/sup&gt; + 20x&lt;/pre&gt;Now, subtract the obtained product from the dividend and write the result immediately below... like this:&lt;pre&gt;  4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5 = -4x ...
-(4x&lt;sup&gt;3&lt;/sup&gt; + 4x&lt;sup&gt;2&lt;/sup&gt; + 20x)
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      - 3x&lt;sup&gt;2&lt;/sup&gt; -  3x - 15&lt;/pre&gt;The last line basically shows what is left after the product between the divisor and the temporary result (-4x) is taken away from the dividend.
The rest of the process repeats itself, with the new line being your new dividend.
So, the question is then &quot;What is (-3x&lt;sup&gt;2&lt;/sup&gt;)/(-x&lt;sup&gt;2&lt;/sup&gt;) and the answer should be 3. So, the next term to add to our temporary result is simply 3.&lt;pre&gt;4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5 = -4x + 3 ...&lt;/pre&gt;So, using the same principles, we now multiply this new result (3) by the divisor and write the result immediately below the new dividend.&lt;pre&gt;  4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5 = -4x + 3 ...
-(4x&lt;sup&gt;3&lt;/sup&gt; + 4x&lt;sup&gt;2&lt;/sup&gt; + 20x)
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      - 3x&lt;sup&gt;2&lt;/sup&gt; -  3x - 15
      - 3x&lt;sup&gt;2&lt;/sup&gt; -  3x - 15&lt;/pre&gt;Subtract the result obtained from the new dividend and write the result immediately below. This will get us:&lt;pre&gt;  4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5 = -4x + 3 ...
-(4x&lt;sup&gt;3&lt;/sup&gt; + 4x&lt;sup&gt;2&lt;/sup&gt; + 20x)
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      - 3x&lt;sup&gt;2&lt;/sup&gt; -  3x - 15
    -(- 3x&lt;sup&gt;2&lt;/sup&gt; -  3x - 15)
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                     0&lt;/pre&gt;This means that when the product between (-4x + 3) and the divisor is taken away from the original dividend, we are left with 0. So, this clearly means that:&lt;pre&gt;4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 = (-x&lt;sup&gt;2&lt;/sup&gt; - x - 5)(-4x + 3).&lt;/pre&gt;We have the result (-4x + 3) and the complete process of obtaining this result is shown once again:&lt;pre&gt;  4x&lt;sup&gt;3&lt;/sup&gt; +  x&lt;sup&gt;2&lt;/sup&gt; + 17x - 15 : -x&lt;sup&gt;2&lt;/sup&gt; - x - 5 = -4x + 3
-(4x&lt;sup&gt;3&lt;/sup&gt; + 4x&lt;sup&gt;2&lt;/sup&gt; + 20x)
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      - 3x&lt;sup&gt;2&lt;/sup&gt; -  3x - 15
    -(- 3x&lt;sup&gt;2&lt;/sup&gt; -  3x - 15)
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                     0&lt;/pre&gt;</PRE>