Explanation

Substituting the given root x = b − 6 into the quadratic gives (b − 6)² − 6(b − 6) + b = 0. Expanding: b² − 12b + 36 − 6b + 36 + b = b² − 17b + 72 = 0, which factors as (b − 8)(b − 9) = 0. Thus b = 8 or b = 9 (both integers, as required), and b² is 64 or 81. The maximum value of b² is therefore 81. Hence option (d). The distractors 36 and 49 are the squares 6² and 7², which would require b = 6 or 7 — values that do not satisfy the substituted equation.