Your math looks fine; the equation of the slant asymptote is y = x + 3.
(There is a typographical error in your factored form; the exponent in the denominator should be 3.)
What do you mean by saying that the slant on your graph is off?
The slope is one; if the scales on both axes are the same, then the graph of y = x + 3 should make a 45-degree angle with the horizontal axis.
Yes, I typed in the wrong equation in my post. When I say the asymptote is off, I mean that the graph in quadrant 1 begins at around x=3 and avoids the asymptote. As , the graph starts to acknowlege the aymptote. Take a look. Could you explain why this happens?