Final answer:
The expression 27x^2 - 75y^2 is factored completely as 3(3x + 5y)(3x - 5y) by using the difference of squares identity and factoring out the greatest common factor.
Step-by-step explanation:
The expression 27x2 - 75y2 is a difference of squares, which can be factored using the identity a2 - b2 = (a + b)(a - b). To factor the expression completely, first factor out the greatest common factor (GCF) of 3, giving us 3(9x2 - 25y2).
Then recognize that 9x2 is a perfect square (3x)2 and 25y2 is a perfect square (5y)2. Applying the identity, we get 3(3x + 5y)(3x - 5y) as the completely factored form of the expression.