To solve this problem, we first simplify the rational expression and then find all the numbers for which the denominator is not zero.
Step 1: Simplifying the Rational Expression
The given rational expression is (y^2 + y - 12) / (y^2 - 2*y - 3).
To simplify a rational expression, we factorize both the numerator and denominator.
The numerator y^2 + y - 12 can be factorized into (y + 4)(y - 3).
The denominator y^2 - 2*y - 3 can be factorized into (y + 1)(y - 3).
So, the rational expression becomes (y + 4)(y - 3) / (y + 1)(y - 3).
The term (y - 3) is common in both the numerator and denominator and can be cancelled.
After cancellation, the simplified expression is (y + 4) / (y + 1).
Step 2: Find the Numbers for Which the Denominator is Not Zero
In the simplified expression, the denominator is (y + 1). For the denominator to not be zero, y ≠ -1.
However, in the original expression, the denominator was (y - 3)(y + 1), so we have to ensure that the values of y don't make this original denominator equal to zero, which gives us another restriction on y, that y ≠ 3.
Thus, the solution to the problem is:
The simplified rational expression is (y + 4)/(y + 1), and the numbers for which the denominator is not zero are y ≠ -1 and y ≠ 3.