To solve the problem involving the binary operation defined on the set of integers, we need to carefully follow the operation's definition and apply it step by step. Let's break it down. ### Step 1: Understand the Operation The operation \( p * q \) is defined as: \[ p * q = pq + p + q \] This means that when we perform the operation \( p * q \), we multiply \( p \) and \( q \), then add \( p \) and \( q \) to the product. ### Step 2: Calculate \( 3 * 4 \) First, we need to calculate \( 3 * 4 \) using the defined operation. 1. Substitute \( p = 3 \) and \( q = 4 \) into the operation: \[ 3 * 4 = 3 \cdot 4 + 3 + 4 \] 2. Calculate \( 3 \cdot 4 \): \[ 3 \cdot 4 = 12 \] 3. Now add \( 3 \) and \( 4 \) to \( 12 \): \[ 12 + 3 + 4 = 12 + 7 = 19 \] So, we find that: \[ 3 * 4 = 19 \] ### Step 3: Calculate \( 2 * (3 * 4) \) Now that we have \( 3 * 4 = 19 \), we need to calculate \( 2 * 19 \). 1. Substitute \( p = 2 \) and \( q = 19 \) into the operation: \[ 2 * 19 = 2 \cdot 19 + 2 + 19 \] 2. Calculate \( 2 \cdot 19 \): \[ 2 \cdot 19 = 38 \] 3. Now add \( 2 \) and \( 19 \) to \( 38 \): \[ 38 + 2 + 19 = 38 + 21 = 59 \] Thus, we find that: \[ 2 * (3 * 4) = 59 \] ### Final Answer The correct option is **A. 59**. ### Explanation of Other Options - **B. 19**: This is the result of \( 3 * 4 \) alone, not the final operation \( 2 * (3 * 4) \). - **C. 67**: This value does not correspond to any calculation based on the defined operation. - **D. 38**: This is the intermediate result of \( 2 * 19 \) before adding \( 2 \) and \( 19 \). ### Summary - The operation \( p * q \) is defined as \( pq + p + q \). - We first calculated \( 3 * 4 \) to get \( 19 \). - Then, we calculated \( 2 * 19 \) to arrive at the final answer of \( 59 \). - The correct answer is **A. 59**. ### Revision Points - Always follow the operation's definition step by step. - Keep track of intermediate results to avoid confusion. - Double-check calculations to ensure accuracy. - Understand why other options do not fit the problem's requirements.