Okay, we are tasked with adding 8124 and 7259. Let's begin!

Step 1: We'll start by adding the digits 4 & 9 in column 1 and get 13.
We'll write down the last digit 3 and carry the 1 to the next column.

Step 2: We'll start by adding the digits 2 & 5 in column 2 and get 8.

Step 3: We'll start by adding the digits 1 & 2 in column 3 and get 3.

Step 4: We'll start by adding the digits 8 & 7 in column 4 and get 15.
We'll write down the last digit 5 and carry the 1 to the next column.

So, 8124 + 7259 = 15383.

