To solve this problem, we first need to determine the base $b$ of the number system. Given that the product of $44_b$ and $11_b$ equals $1034_b$, we'll start by converting the numbers from base $b$ to decimal (base 10).

Step 1: Convert the numbers from base $b$ to decimal

1. Convert $44_b$ to decimal: $44_b = 4b + 4$
2. Convert $11_b$ to decimal: $11_b = b + 1$