You did nothing mathematically wrong. The lady is correct.
However, you should clear of fractions from the beginning, 
not work on each side separately and then cross-multiply 
later.  It just happened to turn out that the denominators
on both sides were x²-4, but in other problems that will not
necessarily be the case.  So clear of denominators.  You
can probably skip the step where the red multiplication
and cancelling is below, because it's just a matter or
multiplying each numerator by the factors in the LCD that
it does not have underneath it.  You will be able to skip
down to 3(x-2)+1(x+2) = 7, just by multiplying each numerator
by the factors of the LCD which each numerator does not have
underneath it.  I put in the extra steps to show what you are
actually doing.

----------------------------------------------



Factor {x²-4) as (x-2)(x+2)



Then get a least common denominator of
(x-2)(x+2) written as 

Then multiply every term by 



Then cancel



3(x-2)+1(x+2) = 7     <--- you could skip the above two steps to here
                           by multiplying each numerator by the factors
                           in the LCD which are not underneath it.

3x-6+x+2 = 7
4x-4 = 7
4x = 11
 x = 

Also there is only one solution.

Edwin