Factor the polynomial completely, or state that the polynomial is prime. Select the correct choice below and fill in any answer boxes within your choice. A. \( 4 x^{4}-324=\square \) B. The polynomial is prime.

To factor the polynomial \( 4x^4 - 324 \), we first notice that both terms share a common factor of 4. Factoring out 4 gives us: \[ 4(x^4 - 81) \] Next, we recognize that \( x^4 - 81 \) is a difference of squares, as it can be rewritten as \( (x^2)^2 - (9)^2 \). We can apply the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \): \[ x^4 - 81 = (x^2 - 9)(x^2 + 9) \] Now, \( x^2 - 9 \) is also a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3) \] The term \( x^2 + 9 \) does not factor further over the real numbers as it has no real roots. Putting it all together, we have: \[ 4(x - 3)(x + 3)(x^2 + 9) \] So, the fully factored form of the polynomial \( 4x^4 - 324 \) is: \[ 4(x - 3)(x + 3)(x^2 + 9) \] If you fill in the answer box, it should be: A. \( 4(x - 3)(x + 3)(x^2 + 9) \)