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

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

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

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

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

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

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

\(\left(a^2+b^2-5\right)^2-4\left(ab+2\right)^2\)

\(=\left(a^2+b^2-5\right)^2-2^2\left(ab+2\right)^2\)

\(=\left(a^2+b^2-5\right)^2-\left(2ab+4\right)^2\)

\(=\left(a^2+b^2-5-2ab-4\right)\left(a^2+b^2-5+2ab+4\right)\)

\(=\left[\left(a^2-2ab+b^2\right)-9\right]\left[\left(a^2+2ab+b^2\right)-1\right]\)

\(=\left[\left(a-b\right)^2-3^2\right]\left[\left(a+b\right)^2-1^2\right]\)

\(=\left(a-b-3\right)\left(a-b+3\right)\left(a-b-1\right)\left(a-b+1\right)\) 

\(\left(x-y+4\right)^2-\left(2x+3y-1\right)^2\)

\(=\left(x-y+4-2x-3y+1\right)\left(x-y+4+2x+3y-1\right)\)

\(=\left(5-x-4y\right)\left(3+3x+2y\right)\) 

1/Cho các số thực dương chứng minh:\(\frac{3\left(a^4+b^4+c^4\right)}{\left(a^2+b^2+c^2\right)^2}+\frac{ab+bc+ca}{a^2+b^2+c^2}\ge2\)

2/Cho a,b dương.Chứng minh:\(\left(\frac{a}{b}+\frac{b}{a}\right)+4\sqrt{2}\frac{a+b}{\sqrt{a^2+b^2}}\ge10\)

3/ Cho các số thực dương. Chứng minh: \(\left(a^2+2bc\right)\left(b^2+2ca\right)\left(c^2+2ab\right)\ge abc\left(a+2b\right)\left(b+2c\right)\left(c+2a\right)\)

Chứng minh các hằng đẳng thức sau :

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

b, \(\left(a^2+b^2\right).\left(c^2+d^2\right)=\left(ac+bd\right)^2+\left(ad-bc\right)^2\)

c, \(\left(ax+b\right)^2+\left(a-bx\right)^2+c^2x^2=\left(a^2+b^2+c^2\right).\left(x^2+1\right)\)

d, \(\dfrac{1}{2}.\left(a+b+c\right).\left[\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\right]=a^3+b^3+c^3-3abc\)

e, \(1000^2+1003^2+1005^2+1006^2=1001^2+1002^2+1004^2+1007^2\)