To factor the expression [tex]\(16 x^4 - 81\)[/tex] completely, we'll go through the following steps:
1. Recognize the difference of squares pattern: The given expression [tex]\(16 x^4 - 81\)[/tex] can be seen as a difference of two squares. Recall that the difference of squares formula is [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
Here, we can rewrite [tex]\(16 x^4 - 81\)[/tex] as:
[tex]\[
(4x^2)^2 - 9^2
\][/tex]
2. Apply the difference of squares formula: The difference of squares formula can be applied to [tex]\( (4x^2)^2 - 9^2 \)[/tex] as follows:
[tex]\[
16 x^4 - 81 = (4x^2 - 9)(4x^2 + 9)
\][/tex]
3. Factor [tex]\(4x^2 - 9\)[/tex] further: Notice that [tex]\(4x^2 - 9\)[/tex] itself is another difference of squares because [tex]\(4x^2\)[/tex] and [tex]\(9\)[/tex] can both be written as squares:
[tex]\[
4x^2 - 9 = (2x)^2 - 3^2
\][/tex]
Applying the difference of squares formula again, we get:
[tex]\[
4x^2 - 9 = (2x - 3)(2x + 3)
\][/tex]
4. Combine all factors: Substituting back into the original expression, we combine all the factors together:
[tex]\[
16 x^4 - 81 = (2x - 3)(2x + 3)(4x^2 + 9)
\][/tex]
So, the completely factored form of [tex]\(16 x^4 - 81\)[/tex] is:
[tex]\[
(2 x-3)(2 x+3)\left(4 x^2+9\right)
\][/tex]
Thus, the correct choice among the options provided is:
[tex]\[
(2 x-3)(2 x+3)\left(4 x^2+9\right)
\][/tex]
