Let's work through the given problem step by step.

You are given matrices $A$, $B$, and $C$, and asked to find $x + y$ based on the equation:

$A + B = C$

The matrices are:

$A = \begin{bmatrix} x - 3 & 2 \\ 3 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} x & 3 \\ 2 & y + 2 \end{bmatrix}, \quad C = \begin{bmatrix} 3 & 5 \\ 5 & 4 \end{bmatrix}$

Now, summing matrices $A$ and $B$:

$A + B = \begin{bmatrix} (x - 3) + x & 2 + 3 \\ 3 + 2 & 5 + (y + 2) \end{bmatrix} = \begin{bmatrix} 2x - 3 & 5 \\ 5 & y + 7 \end{bmatrix}$

We are told that $A + B = C$, so:

$\begin{bmatrix} 2x - 3 & 5 \\ 5 & y + 7 \end{bmatrix} = \begin{bmatrix} 3 & 5 \\ 5 & 4 \end{bmatrix}$

From this, we can compare corresponding elements of the matrices:

1. $2x - 3 = 3$
2. $y + 7 = 4$

Solving for $x$:

$2x - 3 = 3$ $2x = 6$ $x = 3$

Solving for $y$:

$y + 7 = 4$ $y = 4 - 7 = -3$

Now, the question asks for $x + y$:

$x + y = 3 + (-3) = 0$

Thus, the value of $x + y$ is $\boxed{0}$.

Please let me know if you'd like any further details or clarifications.

Here are five related questions to expand your understanding:

1. What happens if you change the matrix $A$ to a different size?
2. How would this problem change if $A + B = 2C$?
3. Can you generalize the process of matrix addition for $n \times n$ matrices?
4. What properties of matrix addition are used in this problem?
5. How would you approach this if you were dealing with 3x3 matrices instead of 2x2?

Tip: Always ensure that matrices are of the same dimensions before performing matrix operations like addition or subtraction.