Here is the simplified expression:

$$5(7y + 2) = 35y + 10$$

To simplify the expression $5(7y + 2)$, you need to distribute the 5 to both terms inside the parentheses. Here are the steps:

$$5(7y + 2) = (5 \cdot 7y) + (5 \cdot 2)$$

1. Multiply $5$ by $7y$:
   $$5 \cdot 7y = 35y$$

2. Multiply $5$ by $2$:
   $$5 \cdot 2 = 10$$

3. Combine the results:
   $$35y + 10$$

So, the simplified expression is $35y + 10$.

$$5(7y + 2) = \boxed{35y + 10}$$