\(P=\frac{a}{\sqrt{\left(b+1\right)\left(b^2-b+1\right)}}+\frac{b}{\sqrt{\left(c+1\right)\left(c^2-c+1\right)}}+\frac{c}{\sqrt{\left(a+1\right)\left(a^2-a+1\right)}}\)

\(\ge\frac{2a}{b^2+2}+\frac{2b}{c^2+2}+\frac{2c}{a^2+2}=\left(a+b+c\right)-\left(\frac{ab^2}{b^2+2}+\frac{bc^2}{c^2+2}+\frac{ca^2}{a^2+2}\right)\)

\(=6-\left(\frac{2ab^2}{b^2+4+b^2}+\frac{2bc^2}{c^2+4+c^2}+\frac{2ca^2}{a^2+4+a^2}\right)\ge6-\left(\frac{2ab}{b+4}+\frac{2bc}{c+4}+\frac{2ca}{a+4}\right)\)

\(=6-\left(2a+2b+2c-\frac{8a}{b+4}-\frac{8b}{c+4}-\frac{8c}{a+4}\right)\)

\(=\frac{8a}{b+4}+\frac{8b}{c+4}+\frac{8c}{a+4}-6=\frac{8a^2}{ab+4a}+\frac{8b^2}{bc+4b}+\frac{8c^2}{ca+4c}-6\)

\(\ge\frac{8\left(a+b+c\right)^2}{\left(ab+bc+ca\right)+4\left(a+b+c\right)}-6\ge\frac{288}{\frac{\left(a+b+c\right)^2}{3}+24}-6=2\)

PTĐTTNT:\(3abc+a^2\left(a-b-c\right)+b^2\left(b-a-c\right)+c^2\left(c-b-a\right)-c\left(b-c\right)\left(a-c\right)\)

\(=3abc+a^3-a^2b-a^2c+b^3-b^2a-b^2c+c^3-c^2b-c^2a-\left(abc-bc^2-c^2a+c^3\right)\)

\(=2abc+a^3-a^2b-a^2c+b^3-b^2c-b^2a\)

\(=\left(a^3+a^2b-a^2c\right)-\left(2a^2b+2ab^2-2abc\right)+\left(ab^2+b^3-b^2c\right)\)

\(=a^2\left(a+b-c\right)-2ab\left(a+b-c\right)+b^2\left(a+b-c\right)\)

\(=\left(a+b-c\right)\left(a^2-2ab+b^2\right)\)

\(=\left(a+b-c\right)\left(a^2-2ab+b^2\right)\)

a)\(\left(b-c\right)^3+\left(c-a\right)^3+\left(a-b\right)^3\)

b)\(\left(x+y\right)^5-x^5-y^5\)

c)\(\left(x^2+y^2\right)^3+\left(z^2-x^2\right)^3-\left(y^2+z^2\right)^3\)

d)\(3abc+a^2\left(a-b-c\right)+b^2\left(b-a-c\right)+c^2\left(c-a-b\right)-c\left(b-c\right)\left(a-c\right)\)

e) 2bc(b+2c)+2ac(c-2a)-2ab(a+2b)-7abc

f)3bc(3b-c)-3ac(3c-a)-3ab(3a+b)+28abc

a) A= \(\left(a+b+c\right)^3+\left(a-b+c\right)^3-6a\left(b+c\right)^2\)

b) B= \(\left(2+1\right)\left(2^2+1\right)\left(2^4+1\right)\left(2^8+1\right)\left(2^{16}+1\right)\)

c) C= \(5\left(2x-1\right)^2+4\left(x-1\right)\left(x+3\right)-2\left(5-3x\right)^2\)

d) D= \(\left(9x-1\right)^2+\left(1-5x\right)^2+2\left(9x-1\right)\left(1-5x\right)\)

e) E= \(\left(2a^2+2a+1\right)\left(2a^2-2a+1\right)-\left(2a^2+1\right)^2\)

Bài 1 :

a) \(\left(6x^2+\frac{1}{3}\right)^2\)

b) \(\left(5x-4y\right)^2\)

c) \(\left(2x^2y-3y^2x\right)^2\)

d) \(\left(5x-3\right).\left(5x+3\right)\)

e) \(\left(-4xy-5\right).\left(5-4xy\right)\)

f) \(\left(a^2b+ab^2\right).\left(ab^2-a^2b\right)\)

g) \(\left(3x-4\right)^2+2.\left(3x-4\right).\left(4-x\right)+\left(4-x\right)^2\)

h) \(\left(a^2+ab+b^2\right).\left(a^2-ab+b^2\right)-\left(a^4+b^4\right)\)