To solve the problem of factoring [tex]\(4x^2 - 9a^4\)[/tex], we need to recognize the expression as a difference of squares.
The difference of squares formula states:
[tex]\[a^2 - b^2 = (a + b)(a - b)\][/tex]
Here, we identify:
[tex]\[a = 2x\][/tex]
[tex]\[b = 3a^2\][/tex]
Then, expressing [tex]\(4x^2 - 9a^4\)[/tex] as a difference of squares, we can rewrite it as:
[tex]\[4x^2 - 9a^4 = (2x)^2 - (3a^2)^2\][/tex]
Now, applying the difference of squares formula:
[tex]\[(2x)^2 - (3a^2)^2 = (2x + 3a^2)(2x - 3a^2)\][/tex]
Thus, [tex]\(4x^2 - 9a^4\)[/tex] factors into:
[tex]\[(2x + 3a^2)(2x - 3a^2)\][/tex]
Therefore, the correct answer is:
[tex]\[
(C) \left(2 x+3 a^2\right)\left(2 x-3 a^2\right)
\][/tex]
