Factorize the terms in the denominator and add the fractions of the numerator:
Expand the bracket in the numerator and collect like terms. Then multiply by the reciprocal of the denominator:
Factorize the numerator of the first factor and cancel out like terms:
Obviously my guess seems to be correct.
By the way: To prevent any wild guesses you should use parantheses. Your problem should have been written like:
[ (2x/x+3) - (x+2/x) / (x^2 + 3x + 2/x^3+2x^2-3x) ] - (6-9x/ x+2) = (2x/(x+3))-((x+2)/x /((x^2+3x+2)/(x^3+2x^2-3x))) )-(6-9x)/(x+2)