Calculate the square root of 225.

We need to calculate $423_5 - 244_5$.
We subtract column by column, borrowing from the left if necessary.

$$\begin{array}{c c c} & 4 & 2 & 3_5 \\ - & 2 & 4 & 4_5 \\ \hline \end{array}$$

Starting from the rightmost column:
$3 - 4$: We cannot subtract 4 from 3. Borrow 1 from the middle digit (2). The borrowed 1 in base 5 is equivalent to 5. So, $3 + 5 = 8$.
$8 - 4 = 4$. The rightmost digit is 4.

Moving to the middle column:
The middle digit was 2, but we borrowed 1, so it becomes 1.
$1 - 4$: We cannot subtract 4 from 1. Borrow 1 from the leftmost digit (4). The borrowed 1 in base 5 is equivalent to 5. So, $1 + 5 = 6$.
$6 - 4 = 2$. The middle digit is 2.

Moving to the leftmost column:
The leftmost digit was 4, but we borrowed 1, so it becomes 3.
$3 - 2 = 1$. The leftmost digit is 1.

So, $423_5 - 244_5 = 124_5$.

To add the results, we need them in the same base. We will convert $124_5$ to base 10.

$$124_5 = (1 \times 5^2) + (2 \times 5^1) + (4 \times 5^0)$$ $$124_5 = (1 \times 25) + (2 \times 5) + (4 \times 1)$$ $$124_5 = 25 + 10 + 4$$ $$124_5 = 39_{10}$$

$$15_{10} + 39_{10} = 54_{10}$$

The simplified expression is 54.

$$\sqrt{225} + (423_5 - 244_5) = 15 + 124_5 = 15 + 39 = \boxed{54}$$