Maximizing $f(x,y) = (a + x)(b + y)$ under the constraint $d=x+y$, where $a$, $b$, $d$ are known. 
Find the maximum of the function $f(x,y) = (a + x)(b + y)$ under the constraint $d = x + y$, where $a$, $b$, $d$ are known.

It seems "obvious" to me that you'd want to split it between the two so that both sides of the product are as equal as possible but no idea how to prove the result.
Sorry I don't know what kind of math this is so I tagged a few.

It's labeled as pre-calculus problem so I'll solve it without calculus.
$f(x,y)=(a+x)(b+y)=(a+x)(b+d-x)=-x^2+(b+d-a)x+(ab+ad)=-(x-(b+d-a)/2)^2+(ab+ad+(b+d-a)^2/4)$
The maximum can be obtained at $x=(b+d-a)/2$ and the maximum value is $ab+ad+(b+d-a)^2/4$
