C1 Optimisation help

I need some help with this question:

"The sum of two positive integers is 40. Find the two integers such that the product of the square of one number and the cube of the other number is a maximum."

I tried to do this question but didn't get very far:

$x + y = 40$
$x = 40 - y$
$P = x^2 \times y^3$
$P = (40-y)(40-y)(y^3)$
$P = (y^2 -80y + 1600)(y^3)$
$P = y^5 - 80y^4 + 1600y^3$

then:

$\frac{dP}{dy} = 5y^4 - 320y^3 + 4800y^2$

Do I need to solve this for 0, or is there an easier way?

$\frac{dP}{dy} = 5y^4 - 320y^3 + 4800y^2$

Do I need to solve this for 0, or is there an easier way?

That all looks fine. Yes, solve dP/dy = 0 which is easier than it looks - you can divide through by 5, and also be y^2, as you are told that y is not zero (because it is positive). You'll get two answers, and you'll have to explain why you reject one of them.