Answer:
(x+4)(x-4)(x+4i)(x-4i) (answer D)
Step-by-step explanation:
We can re-write the original binomial as a difference of squares noticing that [tex]456=16^2[/tex] and that [tex]x^4=(x^2)^2[/tex]
Then we have:
[tex]x^4-256=(x^2)^2-16^2[/tex]
Then we can factor this out using the difference of squares factor form:
[tex](x^2)^2-16^2=(x^2+16)(x^2-16)[/tex]
Now, [tex](x^2-16)[/tex], is itself a difference of squares which we can factor out further:[tex]x^2-16=x^2-4^2=(x+4)(x-4)[/tex]
And we can also solve for the binomial: [tex](x^2+16)[/tex]:
[tex]x^2+16=0\\x^2=-16\\x=+/-\sqrt{-16} \\x=+/-i\,\sqrt{16} \\x=+/-4\,i[/tex]
then we can write [tex](x^2+16)=(x+4i)(x-4i)[/tex]
Therefore, the final factor form of the original binomial is the product of all factors we found: [tex](x+4)(x-4)(x+4i)(x-4i)[/tex]
