Okay, we are tasked with adding 5012 and 2591. Let's begin!

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

Step 2: We'll start by adding the digits 1 & 9 in column 2 and get 10.
We'll write down the last digit 0 and carry the 1 to the next column.

Step 3: We'll start by adding the digits 0 & 5 in column 3 and get 6.

Step 4: We'll start by adding the digits 5 & 2 in column 4 and get 7.

So, 5012 + 2591 = 7603.

