Final answer:
To completely factor the expression 3u^4x^4-48x^4, identify and factor out the greatest common factor, then factor the difference of squares to arrive at the final factored form 3x^4(u^2+4)(u+2)(u-2).
Step-by-step explanation:
To completely factor the expression 3u^4x^4-48x^4, we first look for common factors in both terms. We can see that 3 and x^4 are common to both terms, so we can begin by factoring them out.
This gives us:
3x^4(u^4-16)
Looking at the second term inside the parentheses, u^4-16, we recognize it as a difference of squares because it can be written as u^4-4^2. This allows us to factor it further:
3x^4(u^2+4)(u^2-4)
Again, we have a difference of squares with u^2-4, which can be factored as (u+2)(u-2). Thus, our final factored form of the expression is:
3x^4(u^2+4)(u+2)(u-2)
Remember, when factoring, look for common factors and recognize special products such as a difference of squares.