Which expression is a perfect square of a binomial based on \(a^2-2ab+b^2=(a-b)^2\)?

Correct answer: \(16x^2-24xy+9y^2\)

The correct answer is C. The first term \(16x^2\) is \((4x)^2\), and the last term \(9y^2\) is \((3y)^2\). For the identity \((a-b)^2=a^2-2ab+b^2\), the middle term must be \(-2(4x)(3y)=-24xy\). Option C has exactly this value, so \(16x^2-24xy+9y^2=(4x-3y)^2\). Option A has the wrong final coefficient: \(8y^2\) is not \((3y)^2\). Option B has a positive middle term, which would correspond to addition, but then the middle coefficient would be positive in a different identity. Option D has \(-12xy\), not the required \(-24xy\). The key check is not only the first and last terms; the cross term must be twice the product of their square roots. Memory cue: square roots 4x and 3y give 24xy when doubled, so the correct negative middle term is -24xy.