Polynomial HCF condition: If (x + 2) is the HCF of x^2 + ax + b and x^2 + cx + d, which relation among a, b, c, d must hold?

Introduction / Context:
When (x + 2) is a common factor (and actually the HCF) of two polynomials, both polynomials must vanish at x = −2. Converting that fact to coefficient relations is a standard method to determine constraints among a, b, c, d.

Given Data / Assumptions:

• Polynomials: f(x) = x^2 + ax + b and g(x) = x^2 + cx + d.
• (x + 2) is a common factor ⇒ f(−2) = 0 and g(−2) = 0.

Concept / Approach:
Evaluate each polynomial at x = −2 to get two linear equations: 4 − 2a + b = 0 and 4 − 2c + d = 0. Rearranging yields equalities that can be combined into a single relation matching one of the answer choices.

Step-by-Step Solution:

Verification / Alternative check:
Substitute b = 2a − 4 and d = 2c − 4 into b + 2c − (2a + d); it simplifies to 0 identically, confirming the relation.

Why Other Options Are Wrong:
They do not follow from f(−2) = g(−2) = 0 simultaneously; testing with sample coefficients that satisfy the correct relation will make other proposed relations fail.

Common Pitfalls:
Evaluating at x = +2 instead of −2, or mixing the signs when rearranging the equalities.

Final Answer:
b + 2c = 2a + d