Final answer:
Ethan was correct in identifying 6x^2 as a factor of the expression 6x^4 - 96x^2. The complete factorization includes two additional factors, which are (x + 4) and (x - 4), because the expression can be factored as a difference of squares.
Step-by-step explanation:
To find the other factors of the expression 6x^4 - 96x^2 when 6x^2 is one of the factors, we can begin by factoring out the common term 6x^2 from each term in the expression:
6x^2(x^2 - 16)
The resulting quadratic expression, x^2 - 16, is a difference of squares, which can be further factored as:
(x + 4)(x - 4)
So the complete factorization is:
6x^2(x + 4)(x - 4)
Ethan correctly identified 6x^2 as one of the factors and the other factors are (x + 4) and (x - 4).