so I got
A= -3 , B = 5 C = 5
and i got ∫-3/x-4 + 5x+5/x²+9 dx for my equation
my answer came out to be
I dont know why it is wrong.
Oh yeah that is pretty bad. Use parentheses! You wrote $\displaystyle 3/x-4 + 5x+5/x^2 + 9 $ and integrated it as $\displaystyle 3/(x-4) + 5x+5/(x^2 + 9) $ when in fact you should have written $\displaystyle 3/(x-4) + (5x+5)/(x^2 + 9) $. These expressions are completely different from one another.
Adding to this, as a diagnostic of where you're at could you please do the following two integrals:
$\displaystyle I_1 = \int \frac{2x}{x^2 + 9} \, dx$
$\displaystyle I_2 = \int \frac{5}{x^2 + 9} \, dx$
Edit: If you cannot do these there is little point in you attempting questions like the one you posted until you have thoroughly reviewed the basic integration techniques this questions assumes and have developed a more solid understanding of integration.