To simplify this expression, we first need to find a common denominator for the fractions involved.

Given expression: (2y + 1)/(y^2 + 6y + 9) - (y - 2)/(y^2 - 3y) - y^2 + 6/(y^3 - 9y)

First, factor the denominators:
(y^2 + 6y + 9) = (y + 3)(y + 3)
(y^2 - 3y) = y(y - 3)
(y^3 - 9y) = y(y^2 - 9) = y(y + 3)(y - 3)

Now, rewrite the expression with the common denominator:
[(2y + 1)(y - 3) - (y - 2)(y + 3) - y^2 + 6(y^2 + 3)] / [(y + 3)(y + 3)(y - 3)]

Now, expand and simplify the numerator:
(2y^2 - 5y - 3 - y^2 + 3y - 2 - y^2 + 6y^2 + 18) / [(y + 3)(y + 3)(y - 3)]
(7y^2 + 2y + 13) / [(y + 3)(y + 3)(y - 3)]

Therefore, the simplified expression is (7y^2 + 2y + 13) / [(y + 3)(y + 3)(y - 3)].