Equations and graphs problem

i have an equation
x^2 + 5x +6
--------------
x+2

and get the graph of y=x shifted 3 units up, and when x = -2 there is a one point assymptote.

but when i simpify that equation i get x+3
and the graph is a y=x shifted 3 units up, and no point of assymptote.

so i know both equations are equal to each other, but what would be the correct graph of it since there are two different graphs...???(Thinking)
can someone explain all this....

Subtle.

The functions (not "equations") are not technically equal to each other. The first one does not exist at because is undefined.

If you get onto calculus you'll learn about "limits" and how the "limit" of the first function when x "tends towards" -2 is equal to 1. But the very fact of there being a zero on the bottom at one point means that, as it stands, the functions are indeed not the same.