To factorize 5y^2 + 2ay - 3a^2, we need to find two numbers that multiply to give -15a^2 and add up to 2a. These numbers are 5a and -3a.
We can then rewrite the expression as:
5y^2 + 5ay - 3ay - 3a^2
Factorising by grouping:
(5y^2 + 5ay) - (3ay + 3a^2)
Factorising out the common factor in each group:
5y(y + a) - 3a(y + a)
Factorising out the common factor (y + a):
(5y - 3a)(y + a)
Therefore, the answer is (5y - 3a)(y + a).