9. Which expression can be factored into the form (ax+by)^2 where a and b are real number constants?
Answer: C
C) 4x²+20xy+25y² can be factored into the form (ax+by)².
This expression can be rewritten as (2x + 5y)², indicating that it fits the required format where a and b are real constants.
A) 36x²-49y^2
This expression is a difference of squares and can be factored as (6x - 7y)(6x + 7y), which does not match the form (ax + by)² since it results in two distinct factors rather than a single squared term.
B) 81x²+64y²
While this expression represents a sum of squares, it cannot be factored into the form (ax + by)². The sum of squares cannot be expressed as a perfect square in the realm of real numbers.
C) 4x²+20xy+25y²
This expression can indeed be factored as (2x + 5y)², which directly matches the required form. This shows that both constants a and b are real numbers, satisfying the conditions of the problem.
D) 16x²-24xy-9y²
This expression can be factored into (4x - 3y)(4x + 3y), which again does not fit the format (ax + by)². Like Option A, it results in two separate factors rather than a single squared term.
Conclusion
The expression C) 4x²+20xy+25y² is the only option that can be factored into the form (ax + by)², confirming it as the correct answer. The other options either represent differences of squares, sums of squares, or separate factors, which fail to meet the specified criteria. Thus, C stands out as the sole expression appropriate for the given form.