\( ( x + 2 y ) ( x - 2 y ) ( x ^ { 2 } - 2 x y + 4 y ^ { 2 } ) ( x ^ { 2 } + 2 x y + \Delta y ^ { 2 } ) \)

In the given expression, we have a product of four factors, which can be recognized to relate to specific algebraic identities. The term \( ( x + 2y )( x - 2y ) \) can be simplified using the difference of squares, resulting in \( x^2 - 4y^2 \). The other two factors contain quadratic expressions that can also be analyzed individually to factor further or establish relations, depending on the values assigned to \( \Delta \). When dealing with such polynomial expressions, it's crucial to remember to always check for common factors and relevant identities, like the sum or difference of squares, or perfect square trinomials. Recognizing these patterns can simplify calculations and lead to efficient problem-solving, particularly in mathematics and algebra where these forms frequently occur.