The phrase "the product of the sum of $x$ and $y$ and the difference of $x$ and $y$" translates into the following mathematical expression:

$(x + y)(x - y)$

This is a difference of squares formula, which simplifies to:

$(x + y)(x - y) = x^2 - y^2$

So, the product of the sum and the difference of $x$ and $y$ is $x^2 - y^2$.

Would you like more details on how this works or have any other questions?

Here are 5 related questions to explore further:

1. What happens if $x = 3$ and $y = 2$?
2. How does this formula relate to factoring quadratic expressions?
3. How does this concept apply in solving equations involving squares?
4. What would the result be if both $x$ and $y$ were negative numbers?
5. How can this formula be used to simplify complex algebraic expressions?

Tip: The difference of squares formula can help simplify many algebraic expressions and solve quadratic equations efficiently.