What is the largest possible product for 2 even integers whose sum is 50  ?

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Generally, to maximize a product between two numbers whose sum needs to add up to some specificed number, we want the two numbers to be as close as possible. In this case, it will be 24 and 26.

$$ 24\cdot 26 = \boxed{624} $$

We can treat this problem as a system of equations.

The product:

$$ P=xy $$

A sum of two numbers:

$$ x+y=50 $$

We can use substitution to generate a function.

$$ x= 50-y $$ $$ P = (50-y)y $$ $$ P=50y-y^2 $$

This is a downward facing parabola with vertex at \(y=25\). Since our numbers need to be even integers, the points that result in a maximum value are \((24, 624)\) and \((26, 624)\).