(X+3)/(X+3) is equal to 1. If we multiply X+2 by 1, it should still be X+2. Does this mean X+2 is the same as ((X+2)(X+3))/(X+3)?

They are not exactly the same. They will have the same value for all values for x except -3.

If x = -3, then x + 2 = 1, but ((x+2)(x+3))/(x+3) is undefined because it a denominator of zero, and division by zero is undefined.