Value of (a + b + 2c)² is:

Question

Value of (a + b + 2c)² is:

1. A. a² + b² + 4c² + 2ab + 2bc + 2ca
2. B. a² + b² + 4c² + 2ab + 4bc + 2ca
3. C. a² + b² + 4c² + 2ab + 4bc + 4ca (Correct answer)
4. D. None of the above

Correct Answer

Option C — a² + b² + 4c² + 2ab + 4bc + 4ca

Detailed Solution & Explanation

The correct answer is a² + b² + 4c² + 2ab + 4bc + 4ca.

Key Points

• Apply (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx with x = a, y = b, z = 2c.
• Squares: a² + b² + (2c)² = 4c².
• Cross terms: 2(a)(b) = 2ab, 2(b)(2c) = 4bc, 2(2c)(a) = 4ca.
• Total: a² + b² + 4c² + 2ab + 4bc + 4ca.

Additional Information

• The commonest error is forgetting that the 2 in 2c is squared and also doubles both cross terms containing c — which is exactly how the distractors are built.
• Verify quickly by substituting simple values: with a = b = c = 1, the expression is (1+1+2)² = 16, and the chosen option gives 1 + 1 + 4 + 2 + 4 + 4 = 16 ✓.
• That substitution check works for any identity question and takes only seconds.