Answer:
(4x^2 + 9)(2x + 3)(2x - 3)
Step-by-step explanation:
The first step in factoring is to check for a common factor in all terms. There is no common factor for 16x^4 and 81 other than 1.
Now we look at what we have. We have two terms. The first term, 16x^4, is the square of 4x^2. The second term, 81, is the square of 9. We have a difference of two squares. A difference of two squares factors like this:
a^2 - b^2 = (a + b)(a - b)
We apply that factoring to 16x^4 - 81.
16x^4 - 81 = (4x^2 + 9)(4x^2 - 9)
Factoring means factoring completely, so now we look at what we have, and we see if we can factor it further. We have two binomials with the terms 4x^2 and 9 and 4x^2 and -9. 4x^2 is the square of 2x. 9 is the square of 3. We have a sum of two squares and a difference of two squares. A sum of two squares is not factorable. The difference of two square, 4x^2 - 9, is factorable using the same factoring shown above.
16x^4 - 81 =
= (4x^2 + 9)(4x^2 - 9)
= (4x^2 + 9)(2x + 3)(2x - 3)
Every factor is fully factored, so the answer is:
(4x^2 + 9)(2x + 3)(2x - 3)
