To simplify the rational expression, we can factor out any common terms in the numerator and denominator and then cancel out those common terms.
In the numerator, we have 3x^2y, 9xy^2, and -12y^3. We can factor out y from all of these terms:
y(3x^2 + 9xy - 12y^2)
In the denominator, we have 36x^3y, -27x^2y^2, and -9xy^3. We can factor out 9xy from all of these terms:
9xy(4x^2 - 3xy - y^2)
Now we can rewrite the rational expression factored:
y(3x^2 + 9xy - 12y^2) / (9xy)(4x^2 - 3xy - y^2)
Next, we can cancel out the common factors in the numerator and denominator, which is y and 9xy:
(3x^2 + 9xy - 12y^2) / (4x^2 - 3xy - y^2)
Therefore, the simplified form of the rational expression is (3x^2 + 9xy - 12y^2) / (4x^2 - 3xy - y^2).