Final answer:
Lucas's reasoning is incorrect. The expansion of (a+b)² is actually a² + 2ab + b², not just a² + b², due to the inclusion of the cross-term 2ab from applying the distributive property. Therefore the correct answer is B.)No, Lucas's reasoning is not correct.
Step-by-step explanation:
No, Lucas's reasoning is not correct. The correct expansion of (a+b)² is a² + 2ab + b², not a² + b² as Lucas thinks. This is because when you square a binomial (a sum of two terms), you need to apply the distributive property, which involves squaring both terms individually and adding the product of the two terms multiplied together. Here's the step-by-step expansion:
(a+b)² = (a+b) × (a+b)(a+b)² = a² + ab + ab + b²(a+b)² = a² + 2ab + b²
The mistake made by Lucas is overlooking the cross-terms (in this case, 2ab). This is a common error and serves as an important reminder that the distributive property must be fully applied when squaring binomials.