Okay, we are given 5258 and 5629. Let's add them up step by step.

Step 1: We'll start by adding the digits 8 & 9 in column 1 and get 17.
We'll write down the last digit 7 and carry the 1 to the next column.

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

Step 3: We'll start by adding the digits 2 & 6 in column 3 and get 8.

Step 4: We'll start by adding the digits 5 & 5 in column 4 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

So, 5258 + 5629 = 10887.

