at what price he should sell his product? Can anyone help me with this problem?

A merchant has determined that if the selling price of peaches is 15 cent each, he will sell 400 of them each day. He has also determined that for each cent the price is increased he will sell 16 fewer peaches each day. At what price in cents should he sell each peach in order to maximize his income from peaches each day?

Here is what I did: assume y is the total price and x is the amount of cent increase. Then I got: $$y=(15+x)(400-16x)=-16x^2+(400-15*16)x+400*15=-16x^2+160x+16*25*15=-16[x^2-10x-25*15]$$ Then how do I determine what x make y largest?

As you have correctly deduced, we have $P(n) = (15 + n)(400 - 16n) = -16n^2 + 160n + 6000$
At this point, you want to complete the square so that we have 
$$P(n) = -16n^2 + 160n + 6000 = -16(n^2 - 10n) + 6000 = -16(n^2 - 10n + 25) + 6400 = -16(n-5)^2 + 6400$$
Is the expression on the right-most side of the above equation easier to maximize? 
Remark: In this case, we're lucky since the maximum of $P(n)$ occurs at a natural number. Otherwise, assuming it is a parabola, you would consider $P$ with domain $\mathbb{R}$ and, at whatever value $c$ its maximum occurs, choose $n = [c]$ where $[x]$ is the nearest integer function. 
