Question 1203479
Depends what you are expected to know or what method you are told to use.

One learns very early on {{{(a-b)(a+b)=a^2+ab-ab+b^2=a^2-b^2}}}


Seeing you have the difference of two squares,
{{{4x^2-9}}}

{{{(2x)^2-3^2}}}

{{{((2x)+3)((2x)-3)}}}
So the division to perform using this is obvious.



Maybe you want to use polynomial division?
<pre>
           2x     -3
       ____________________________
2x+3   |   4x^2    0x    -9
           4x^2    6x
         --------------
           0      -6x   -9
                  -6x   -9
                 ---------
                  0    0     remainder 0
</pre>
Directly demonstrating  {{{4x^2-9}}} divided by {{{2x+3}}}  is  {{{highlight(2x-3)}}}.