Given $x = 2 + 3\sqrt{2}$ and $y = -1 + 5\sqrt{2}$, find the value of $x^2 - y^2$.

1. Problem Description

Given $x = 2 + 3\sqrt{2}$ and $y = -1 + 5\sqrt{2}$, find the value of $x^2 - y^2$.

2. Solution Steps

We are given that $x = 2 + 3\sqrt{2}$ and $y = -1 + 5\sqrt{2}$.

We need to find $x^2 - y^2$. We can factorize the expression as $(x-y)(x+y)$.

First, let's find $x+y$:

$x+y = (2 + 3\sqrt{2}) + (-1 + 5\sqrt{2}) = 2 - 1 + 3\sqrt{2} + 5\sqrt{2} = 1 + 8\sqrt{2}$

Next, let's find $x-y$:

$x-y = (2 + 3\sqrt{2}) - (-1 + 5\sqrt{2}) = 2 + 3\sqrt{2} + 1 - 5\sqrt{2} = 3 - 2\sqrt{2}$

Now, we can find $x^2 - y^2 = (x+y)(x-y)$:

$x^2 - y^2 = (1 + 8\sqrt{2})(3 - 2\sqrt{2})$

$= 1(3) + 1(-2\sqrt{2}) + 8\sqrt{2}(3) + 8\sqrt{2}(-2\sqrt{2})$

$= 3 - 2\sqrt{2} + 24\sqrt{2} - 16(2)$

$= 3 + 22\sqrt{2} - 32$

$= -29 + 22\sqrt{2}$

Therefore, $x^2 - y^2 = -29 + 22\sqrt{2}$.

3. Final Answer

$-29 + 22\sqrt{2}$