coefficient

There are 3 ways to get a $y^4$ term: $(2y)^4$, $(3y^2) ^2$, and $(2y)^2 \times (3y^2)$

For the first way, we need 4 $2y$'s  and 2 $-5$'s, so we have$(2y)^4 \times (-5)^2 = 400y^4$. But we also need to multiply by ${6 \choose 2} = 15$ because there are 15 ways to order them. This makes for $400y^4 \times 15 = 6000y^4$

For the second way, we need 2  $3y^2$'s and 4 $-5$'s, so we have $(3y^2)^2 \times (-5)^4 = 5625y^4$. But like last time, we need to multiply by 15 for the same reason. Thus we have $5625y^4 \times 15 = 84375y^4$. 

For the final way, we need 2 $2y$'s, 1 $3y^2$'s, and 3 $-5$'s, so we have $(2y)^2 \times (3y^2) \times (-5)^3 = -1500y^3$. But this time, we need to multiply by ${6! \over 3!2!} = 60$. Thus we have $-1500y^4 \times 60 = -90000y^4$

So in total, the coefficient of $y^4$ is $6,000y^4 + 84,375y^4 - 90,000y^4 = \color{brown}\boxed{375y^4}$