Simplify the algebraic expression (4a^2 + 8b + 14c + 2) divided by 2 and find the resulting expression in terms of a, b and c.

Introduction / Context:
Algebraic simplification is a basic but vital skill for aptitude and competitive exams. This question asks you to simplify an expression where a polynomial in a, b and c is divided by a constant. Understanding how to distribute division over addition and how to reduce coefficients is key to quickly arriving at the correct simplified form. The task is purely algebraic and involves no substitution or solving for variables.

Given Data / Assumptions:

• The original expression is (4a^2 + 8b + 14c + 2) / 2.
• a, b and c are algebraic variables.
• The division by 2 applies to every term inside the parentheses.
• We assume standard arithmetic rules for real numbers.

Concept / Approach:
To simplify a sum divided by a constant, we divide each term in the sum by that constant. This uses the distributive property: (x + y + z) / k = x / k + y / k + z / k when k is nonzero. After distributing the division, we simplify each coefficient by dividing it by 2. Because all coefficients are even in this problem, the simplification is very straightforward and results in integer coefficients in the final expression.

Step-by-Step Solution:
Step 1: Write the expression as (4a^2 + 8b + 14c + 2) / 2. Step 2: Distribute the division across the sum: 4a^2 / 2 + 8b / 2 + 14c / 2 + 2 / 2. Step 3: Simplify each term: 4a^2 / 2 = 2a^2, 8b / 2 = 4b, 14c / 2 = 7c, and 2 / 2 = 1. Step 4: Combine the simplified terms to form the final expression: 2a^2 + 4b + 7c + 1.

Verification / Alternative check:
You can reverse the process to verify the answer. Multiply the final result 2a^2 + 4b + 7c + 1 by 2, and you should recover the original numerator. Doing this gives 4a^2 + 8b + 14c + 2, which matches the original expression in the numerator. This confirms that the simplification step was applied correctly and that the final expression is accurate.

Why Other Options Are Wrong:

• Option b (a^2 + 4b + 7c + 1) divides only the coefficient of a^2 by 4 instead of 2, which is incorrect.
• Option c (2a^2 + 4b + 7c + 2) keeps the constant term as 2 instead of dividing it by 2 to get 1.
• Option d (a^2 + 4b + 7c + 2) has both the a^2 coefficient and the constant term incorrectly simplified.
• Option e (2a^2 + 8b + 14c + 1) does not reduce the coefficients of b and c at all, so it does not represent the effect of dividing by 2.

Common Pitfalls:
A common mistake is to divide only the first term by the denominator and forget to divide the rest, or to incorrectly handle the constant term. Some students also try to factor the numerator first and then cancel, which can introduce errors if not done carefully. The safest method is usually to distribute the division to each term and simplify coefficients one by one. Clear writing and stepwise simplification help avoid small arithmetic slips that could lead to the wrong option being chosen.

Final Answer:
The simplified form of (4a^2 + 8b + 14c + 2) / 2 is 2a^2 + 4b + 7c + 1.