Final answer:
To reduce the rational expression to lowest terms, factor the numerator and denominator, then cancel out common factors to simplify. The final simplified expression is (3q + 5)/(q - 6).
Step-by-step explanation:
To reduce the rational expression to lowest terms, we need to find the greatest common factor (GCF) of the numerator and the denominator. In this case, the numerator is 3q² + 21q + 30 and the denominator is 9q² − 18q − 72. We can factor each expression to simplify.
The numerator can be factored as (3q + 5)(q + 6) and the denominator can be factored as 9(q - 6)(q + 2). Now, we can cancel out common factors to simplify the expression.
The final simplified expression is (3q + 5)/(q - 6).