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\(a\left(b^2+c^2\right)+b\left(a^2+c^2\right)+c\left(a^2+b^2\right)-2abc-a^3-b^3-c^3\)
\(=c\left(a-b\right)^2+\left[ab^2+ac^2+a^2b+bc^2-a^3-b^3-c^3\right]\)
\(=c\left(a-b\right)^2+c^2\left(a+b-c\right)+ab^2+a^2b-a^3-b^3\)
\(=c\left(a-b\right)^2+c^2\left(a+b-c\right)-\left(a^3-a^2b\right)+\left(ab^2-b^3\right)\)
\(=c\left(a-b\right)^2+c^2\left(a+b-c\right)-a^2\left(a-b\right)+b^2\left(a-b\right)\)
\(=c\left(a-b\right)^2+c^2\left(a+b-c\right)-\left(a+b\right)\left(a-b\right)^2\)
\(=-\left(a-b\right)^2\left(a+b-c\right)+c^2\left(a+b-c\right)\)
\(=\left(a+b-c\right)\left(a-b+c\right)\left(-a+b+c\right)\)
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.\(a\left(b^2+c^2\right)+b\left(c^2+a^2\right)+c\left(a^2+b^2\right)-2abc-a^3-b^3-c^3\)
=\(a\left(b^2-2bc+c^2-a^2\right)+b\left(a^2+2ac+c^2-b^2\right)+c\left(a^2-2ab+b^2-c^2\right)\)
=\(a\left[\left(b-c\right)^2-a^2\right]+b\left[\left(a+c\right)^2-b^2\right]+=c\left[\left(a-b^2\right)-c^2\right]\)
=\(a\left(c-b+a\right)\left(a+b-c\right)+b\left(a+c-b\right)\left(a+b+c\right)+c\left(a-b+c\right)\left(a-b-c\right)\)
=\(\left(a+c-b\right)\left[a\left(c-b+a\right)+b\left(a+b+c\right)+c\left(a-b-c\right)\right]\)
=\(\left(a+c-b\right)\left(b+a-c\right)\left(c+b-a\right)\)
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\(N=a^3+b^3+3ab\)
\(=\left(a+b\right)^3-3ab\left(a+b\right)+3ab\)
=1
\(M=\left(a^2+b^2+2-a^2-b^2+2\right)\left[\left(a^2+b^2+2\right)^2+\left(a^2+b^2+2\right)\left(a^2+b^2-2\right)+\left(a^2+b^2-2\right)^2\right]-12\left(a^2+b^2\right)^2\\ M=4\left(a^4+b^4+4+4a^2+4b^2+2a^2b^2+\left(a^2+b^2\right)^2-4+a^4+b^4+4-4a^2-4b^2+2a^2b^2\right)-12\left(a^4+2a^2b^2+b^4\right)\\ M=4\left(3a^4+3b^4+4+6a^2b^2\right)-12\left(a^4+2a^2b^2+b^4\right)\\ M=4\left(3a^4+3b^4+4+6a^2b^2-3a^4-6a^2b^2-3b^4\right)\\ M=4\cdot4=164\)