Which expression can be factored into the form (ax+by)^2 where a and b are real number constants?

Rationale

4x²+20xy+25y² can be factored into the form (ax+by)².

This expression is a perfect square trinomial, which can be rewritten as (2x + 5y)², demonstrating its ability to fit the required format with real number constants a and b.

A (36x²-49y^2) YOUR ANSWER

This expression represents a difference of squares, which factors into (6x - 7y)(6x + 7y). It cannot be expressed in the form (ax + by)² because it does not result in a perfect square trinomial.

B (81x²+64y^2) YOUR ANSWER

This expression is a sum of squares, which cannot be factored over the reals. The form (ax + by)² requires the expression to be a perfect square, but since it is a sum, it does not meet the criteria for factoring in this way.

C (4x²+20xy+25y^2) CORRECT

This expression can be factored as (2x + 5y)². It is a perfect square trinomial, confirming that it can indeed be written in the desired form of (ax + by)² with a = 2 and b = 5.

D (16x²-24xy-9y^2) YOUR ANSWER

This expression does not factor into a perfect square. Instead, it can be factored into (4x - 3y)(4x + 3y) or using the quadratic formula. Therefore, it cannot be expressed in the form (ax + by)².

Conclusion

Among the given options, only 4x² + 20xy + 25y² is a perfect square trinomial that can be factored into the form (ax + by)². The other choices either represent sums or differences of squares or do not conform to the perfect square structure, confirming that option C is the only correct answer.

Which expression can be factored into the form (ax+by)^2 where a and b are real number constants?

4x²+20xy+25y² can be factored into the form (ax+by)².

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