The problem asks to find the length of side $AB$ of a rectangle where the perimeter is 126 units. The lengths of the sides are given as $AB = 3y + 3$ and $AD = 2y$.

Steps to solve:

1. Perimeter formula of a rectangle: $P = 2 \times (\text{Length} + \text{Width})$ Substituting the given expressions for the length and width: $P = 2 \times \left( (3y + 3) + 2y \right)$ Given that the perimeter $P = 126$, we can set up the equation: $126 = 2 \times \left( (3y + 3) + 2y \right)$

2. Simplify and solve for $y$: $126 = 2 \times (5y + 3)$ Divide both sides by 2: $63 = 5y + 3$ Subtract 3 from both sides: $60 = 5y$ Solve for $y$: $y = 12$

3. Find the length of $AB$: Since $AB = 3y + 3$, substitute $y = 12$: $AB = 3(12) + 3 = 36 + 3 = 39$

Thus, the length of side $AB$ is 39 units.

Would you like more details on this solution? Here are some additional questions you might want to explore:

1. How do you derive the perimeter formula for a rectangle?
2. What are other ways to represent the relationship between the perimeter and the side lengths?
3. How would you verify your answer by recalculating the perimeter?
4. How does this problem change if the rectangle's perimeter was different?
5. Can you generalize this problem for a rectangle with variable side lengths?

Tip: Always double-check your solution by plugging your final answer back into the original equation to verify if the conditions are satisfied!