Suppose $x$ and $y$ are real numbers satisfying $x+y=5$. What is the largest possible value of $x^{2}+2 x y$ ?
The quantity in question is $(x+y)^{2}-y^{2} \leq(x+y)^{2}=25$. Equality occurs when $x=5$ and $y=0$, hence the maximum possible value is 25 .