<p>If the expression \(x^2 + 2(a + b + c)x + 3(bc + ca + ab)\) is a perfect square, then</p>
Step-by-Step Solution
Key Concept: For a quadratic expression to be a perfect square trinomial, its discriminant must equal zero. This means the coefficient constraint 4(a+b+c)² - 12(bc+ca+ab) = 0 must hold, which simplifies to (a+b+c)² = 3(bc+ca+ab).
<p><strong>Step 1:</strong> For the expression x² + 2(a+b+c)x + 3(bc+ca+ab) to be a perfect square, its discriminant must equal zero.</p><p><strong>Step 2:</strong> Calculate discriminant: D = [2(a+b+c)]² - 4(1)(3(bc+ca+ab)) = 0</p><p><strong>Step 3:</strong> Simplify: 4(a+b+c)² - 12(bc+ca+ab) = 0</p><p><strong>Step 4:</strong> Divide by 4: (a+b+c)² - 3(bc+ca+ab) = 0</p><p><strong>Step 5:</strong> Expand (a+b+c)² = a² + b² + c² + 2(ab+bc+ca)</p><p><strong>Step 6:</strong> Substitute: a² + b² + c² + 2(ab+bc+ca) = 3(bc+ca+ab)</p><p><strong>Step 7:</strong> Rearrange: a² + b² + c² = ab + bc + ca</p><p><strong>Step 8:</strong> Multiply by 2: 2a² + 2b² + 2c² = 2ab + 2bc + 2ca</p><p><strong>Step 9:</strong> Recognize: (a-b)² + (b-c)² + (c-a)² = 0, which implies a = b = c</p><p>∴ Answer: A</p>
Correct Answer: A