Question 278156
{{{(8x^3-4x^2-6x-36)/(x-4)}}}
Since this fraction does not simplify. I'm guessing that the actual fraction is:
{{{(8x^3-4x^2-6x-36)/(x^2-4)}}}<br>
Simplifying fractions involves canceling factors that are common to the nnumerator and denominator. To cancel factors we need factors.<br>
So we start by factoring. The numerator has a GCF of 2 which we can factor out. And the denominator is a difference of squares so it factors easily.
{{{(2(4x^3-2x^2-3x-18))/((x+2)(x-2))}}}
We don't have common factors yet so we keep factoring. 
{{{(2(4x^3-2x^2-3x-18))/((x+2)(x-2))}}}
The second factor in the numerator<ul><li>doesn't fit any of the factoring patterns</li><li>has too many terms for trinomial factoring</li><li>doesn't appear to be factorable by grouping.</li></ul>
So it seem that we need to factor by trial and error of the possible rational roots. The possible rational roots are the ratios, positive and negative, which can be formed using a factor of the constant term (at the end, 18 in your case) over a factor of the leading coefficient, 4. The factors of 18 are 1, 2, 3, 6, 9 and 18. The factors of 4 are 1, 2 and 4. So there are a lot of possible rational roots. But we're only interested in factors that match a factor in the denominator. So the only roots worth trying here are 2 and -2. I'm going to try 2 first. To check a rational root is probably easiest with synthetic division:
<pre>
2 |  4  -2  -3  -18
         8  12   18
    ---------------
     4  -6   9    0
</pre>
The remainder is zero so 2 is a rational root and (x-2) is a factor. And, from the numbers in front of the remainder, the other factor is {{{4x^2 -6x + 9}}}. So now our factored fraction is:
{{{(2(x-2)(4x^-6x+9))/((x+2)(x-2))}}}
We could try to continue to factor the numerator but we can see that the rational roots of {{{4x^2 -6x + 9}}} do not include -2. So x+2 will not be a factor. Since any other factors do not help we won't bother factoring any more. Now we can cancel the common factor of x-2:
{{{(2(4x^2-6x+9))/(x+2)}}}
which simplifies to:
{{{(8x^2-12x+18)/(x+2)}}}