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With these messy coefficients, I use the quadratic equation to find the roots.
For the numerator we have
a = 9
b = 40y
c = -15y^2
Using (-b+/-sqrt(b^2-4ac))/2a, we obtain the roots of the numerator are
(1) x = 5/9y, -5y which gives the factors of the numerator as
(3) Num = 9(x-5/9y)(x+5y) or
(4) Num = (9x-5y)(x+5y)
Foil (4) to check that
Num = 9x^2 +40xy -25y^2 as given.
Do the same for the denominator, ie use the quadratic equation with
a = 18
b = 17y
c = -25y^2 which gives the roots
(5) x = 5/9y, -3/2y which gives the factors of the denominator as
(6) Den = 18(x-5/9y)(x+3/2y) which reduces to
(7) Den = (9x-5y)(2x+3y)
FOIL (7) to check that
Den = 18x^2 +17xy -15y^2 as given.
The given expression is
(8) Num/Den = (4)/(7) = [(9x-5y)(x+5y)]/[(9x-5y)(2x+3y)]
Simplifying (8) yields the answer