Answer :

Sure! Let's factor the expression [tex]\(27x^2 - 12y^2\)[/tex] step by step.

1. Identify a Common Factor:
Start by identifying the greatest common factor (GCF) of the coefficients [tex]\(27\)[/tex] and [tex]\(12\)[/tex].
- The GCF of [tex]\(27\)[/tex] and [tex]\(12\)[/tex] is [tex]\(3\)[/tex].

2. Factor Out the GCF:
Factor out [tex]\(3\)[/tex] from the expression:
[tex]\[
27x^2 - 12y^2 = 3(9x^2 - 4y^2)
\][/tex]

3. Recognize as a Difference of Squares:
[tex]\(9x^2 - 4y^2\)[/tex] is a difference of squares. Recall that a difference of squares can be factored using the formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
Here, [tex]\(a^2 = 9x^2\)[/tex] and [tex]\(b^2 = 4y^2\)[/tex], so [tex]\(a = 3x\)[/tex] and [tex]\(b = 2y\)[/tex].

4. Apply the Difference of Squares Formula:
Substitute [tex]\(a = 3x\)[/tex] and [tex]\(b = 2y\)[/tex] into the formula:
[tex]\[
9x^2 - 4y^2 = (3x - 2y)(3x + 2y)
\][/tex]

5. Combine with the Factored Out GCF:
Put it all together with the GCF factored out earlier:
[tex]\[
27x^2 - 12y^2 = 3(3x - 2y)(3x + 2y)
\][/tex]

Thus, the fully factored form of [tex]\(27x^2 - 12y^2\)[/tex] is:
[tex]\[
3(3x - 2y)(3x + 2y)
\][/tex]

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