Please note that -4i/2-3i means , not 
You MUST put parentheses around any numerator or denominator
which contains more than one number or one letter to tell
where it starts and ends.  I am quite sure you meant 
and not .  You should have typed -4i/(2-3i). 

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Multiply by the conjugate of the denominator
written over itself:



Indicate the multiplication of the numerators and denominators:



Distribute the top and FOIL the bottom:



Cancel the middle terms in the bottom:





Since  substitute (-1) for i2.



Simplify the denominator:





Write as the sum of two fractions:



Reverse the terms so the i term will be on the right:



Change the division of 8i by 13 as a multiplication
of i by the fraction :



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Here's an easy way if you have a TI-83 or TI-84:

Make sure the calculator is in the mode to take
complex numbers.  You can leave it set that way
permanently, since it doesn't affect any of the
other functions.  Press "MODE", scroll down to
"REAL". Press the right arrow once to put the cursor
on "a+bi" and press "ENTER", then press "2ND" "MODE".
Now it is set to take complex numbers.  You'll never
have to do that again.  The reason the TI company
doesn't put it in that mode at the factory is because
some people, like engineers, have hundreds and hundreds
of stored programs in their calculator and the "a+bi" 
mode takes up a little more memory that those people
might need for their programs. 

Now

Press CLEAR
Press -4i/(2-3i)   <-- the "i" is "2ND" then the decimal key. 
                  (the period or decimal key which is the 
                  middle key at the bottom of the keyboard)
                  Be sure to put those parentheses around
                  that denominator as you mistakenly forgot 
                  them above.

Press 2ND ENTER ENTER 

read 12/13-8/13i

Edwin