Question

1. $12x^2 - 27y^2$

Explanation:

Step1: Factor out the GCF

First, find the greatest common factor (GCF) of \(12x^{2}\) and \(27y^{2}\). The GCF of 12 and 27 is 3. So we factor out 3:
\(12x^{2}-27y^{2}=3(4x^{2}-9y^{2})\)

Step2: Apply the difference of squares formula

Notice that \(4x^{2}-9y^{2}\) is a difference of squares, which can be factored using the formula \(a^{2}-b^{2}=(a + b)(a - b)\). Here, \(a = 2x\) (since \((2x)^{2}=4x^{2}\)) and \(b = 3y\) (since \((3y)^{2}=9y^{2}\)). So we factor \(4x^{2}-9y^{2}\) as:
\(4x^{2}-9y^{2}=(2x + 3y)(2x - 3y)\)

Step3: Combine the factors

Substitute the factored form of \(4x^{2}-9y^{2}\) back into the expression from Step 1:
\(3(4x^{2}-9y^{2})=3(2x + 3y)(2x - 3y)\)

Answer:

\(3(2x + 3y)(2x - 3y)\)