\( x ^ { 2 } + 2 \sqrt { 2 } x y + 2 y ^ { 2 } \)

This expression can be factored as a perfect square: \( (x + \sqrt{2}y)^2 \). This factoring is helpful because it simplifies the process of solving equations or analyzing the behavior of the function represented by the expression. Understanding how to recognize and factor perfect squares can make tackling quadratic forms much easier, like spotting treasure in a mathematical treasure hunt! In terms of real-world application, this expression could represent a variety of scenarios such as modeling projectile motion where \( x \) and \( y \) are variables correlated to time and distance in physics problems, or it could pop up in optimization problems, such as maximizing the area of a garden plot. Knowing how to manipulate such expressions allows for more efficient problem-solving in these situations!