Okay, we are tasked with adding 3365 and 1129. Let's begin!

Step 1: We'll start by adding the digits 5 & 9 in column 1 and get 14.
We'll write down the last digit 4 and carry the 1 to the next column.

Step 2: We'll start by adding the digits 6 & 2 in column 2 and get 9.

Step 3: We'll start by adding the digits 3 & 1 in column 3 and get 4.

Step 4: We'll start by adding the digits 3 & 1 in column 4 and get 4.

So, 3365 + 1129 = 4494.

