Please don't be tempted by the temptation to expand the second term as it will lead to headache and no closer to the answer of the more simplified form:
$2(x^8+y^8+z^8)-(x^4+y^4+z^4)^2$
$=2x^8+2y^8+2z^8-((x^4+y^4)^2+2(x^4+y^4)z^4+z^8)$
$=2x^8+2y^8+2z^8-(x^8+2x^4y^4+y^8+2x^4z^4+2y^4z^4+z^8)$
$=2x^8+2y^8+2z^8-(x^8+y^8+z^8+2x^4y^4+2x^4z^4+2y^4z^4)$
$=x^8+y^8+z^8-2x^4y^4-2x^4z^4-2y^4z^4$
$=x^8-2x^4y^4+y^8-2y^4z^4+z^8-2x^4z^4$
$=x^4(x^4-2y^4)+y^4(y^4-2z^4)+z^4(z^4-2x^4)$
Now, up to this point, there is nothing we could do to simplify the expression, and counting the number of terms we have, we have it one more term than the beginning expression, and we're not simplifying but complicating the given expression. So, we need to stop pursuing from this angle. Take a step back and a deep breath, then making the attempt next by thinking differently and therefore using the different approach now.
As usual, I encourage you to take a deep look at the problem, and tell me what alternative(s) you could consider to assist you to simplify the problem beautifully.
I will post the full solution tomorrow, and I will be happy to hear from you to guide you through the problem if you wanted to, i.e. by commenting or querying at this blog post comment area.
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