Okay, we are tasked with adding 735 and 3772. Let's begin!

Step 1: We'll start by adding the digits 5 & 2 in column 1 and get 7.

Step 2: We'll start by adding the digits 3 & 7 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 7 & 7 in column 3 and get 15.
We'll write down the last digit 5 and carry the 1 to the next column.

Step 4: We'll start by adding the digits 0 & 3 in column 4 and get 4.

So, 735 + 3772 = 4507.

