What is the correct expansion of ( (a-b)^2 )?

Correct answer: (a^2-2ab+b^2)

The direct answer is option B: \\(a^2-2ab+b^2\\). Begin with the meaning of a square: \\(a-b\\)^2 means \\(a-b\\)(a-b)\\). Multiply each term in the first bracket by each term in the second: \\(a\\cdot a=a^2\\), \\(a\\cdot(-b)=-ab\\), \\((-b)\\cdot a=-ab\\), and \\((-b)\\cdot(-b)=b^2\\). Adding these gives \\(a^2-ab-ab+b^2=a^2-2ab+b^2\\). Option B is correct. Option A leaves out the two cross-products, so it is incomplete. Option C has a positive middle term and belongs to \\(a+b\\)^2\\), not \\(a-b\\)^2\\). Option D is the difference of squares, \\(a^2-b^2=(a-b)(a+b)\\), not the square of a difference. The common mistake is forgetting that two negative cross-products combine to \\(-2ab\\).