Question 236721
You were correct in rewriting the numerator. Here's long division with polynomials:
<pre>
     4x^2  + (-4x)  +   6
     ________________________________
2x+3/8x^3  +  4x^2  +   0x  +    7
     8x^3  + 12x^2
     -------------
             -8x^2  +   0x
             -8x^2  + -12x
             -------------
                       12x   +   7
                       12x   +  18
                       -----------
                               -11
</pre>
So
{{{(8x^3+4x^2+7)/(2x+3) = 4x^2 + (-4x) + 6 + (-11)/(2x+3)}}}