If the first and last terms of a trinomial are perfect squares, what condition must its middle term satisfy for the trinomial to be a perfect-square trinomial?

Correct answer: The middle term must be twice the product of the square roots of the first and last terms, with either appropriate sign.

Correct answer: A. If the outer terms are a² and b², a perfect-square trinomial must have the form a² + 2ab + b² = (a + b)² or a² − 2ab + b² = (a − b)². Thus the middle term must be ±2ab: twice the product of the square roots of the first and last terms, with its sign determining whether the binomial has a plus or minus sign. Option B is wrong because the middle term is not generally the sum of the outer terms. Option C is wrong because multiplying the outer terms gives a⁴b⁴ in symbolic form, not the required cross term. Option D is wrong because the middle coefficient can be many values, depending on a and b; it is the relationship ±2ab that matters. Memory cue: outer squares alone are not enough—always test the middle term against ±2ab.