Let's solve the problem step-by-step using the given information. The binary operation is defined on the set of integers p and q by pq = pq + p + q. We need to find the value of 2 (3 4).

First, let's evaluate the expression inside the parentheses, (3 4). According to the definition of the binary operation , we have:

\(3 4 = 3 \cdot 4 + 3 + 4 = 12 + 3 + 4 = 19\)

Now, we can substitute this value into the original expression to get:

\(2 (3 4) = 2 19\)

Using the definition of the binary operation again, we have:

\(2 19 = 2 \cdot 19 + 2 + 19 = 38 + 2 + 19 = 59\)

So, the value of 2 (3 4) is 59.