a) \(\left(x+1\right)^4+\left(x^2+x+1\right)^2\)
\(=\left(x+1\right)^4+\left[x\left(x+1\right)+1\right]^2\)
\(=\left(x+1\right)^4+x^2\left(x+1\right)^2+2x\left(x+1\right)+1\)
\(=\left(x+1\right)^2\left[\left(x+1\right)^2+x^2\right]+\left(2x^2+2x+1\right)\)
\(=\left(x+1\right)^2\left(2x^2+2x+1\right)+\left(2x^2+2x+1\right)\)
\(=\left(2x^2+2x+1\right)\left[\left(x+1\right)^2+1\right]\)
\(=\left(2x^2+2x+1\right)\left(x^2+2x+2\right)\)
b) \(\left(a+b-2c\right)^3+\left(b+c-2a\right)^3+\left(c+a-2b\right)^3\)
Đặt \(x=a+b-2c\)
\(y=b+c-2a\)
\(z=c+a-2b\)
\(\Rightarrow x+y+z=a+b-2c+b+c-2a+c+a-2b\)
\(\Rightarrow x+y+z=0\)
\(\Rightarrow x+y=-z\left(1\right)\)
\(\Rightarrow\left(x+y\right)^3=\left(-z\right)^3\)
\(\Rightarrow x^3+y^3+3xy\left(x+y\right)=\left(-z\right)^3\)
\(\Rightarrow x^3+y^3+z^3+3xy.\left(-z\right)=0\) ( Vì x + y = -z )
\(\Rightarrow x^3+y^3+z^3-3xyz=0\)
\(\Rightarrow x^3+y^3+z^3=3xyz\)
\(\Rightarrow\left(a+b-2c\right)^3+\left(b+c-2a\right)^3+\left(c+a-2b\right)^3=3\left(a+b-2c\right)\left(b+c-2a\right)\left(c+a-2b\right)\)
c) \(\left(x^2-x+2\right)^2-\left(x-2\right)^2\)
\(=x^4+x^2+4-2x^3-4x+4x^2+x^2-4x+4\)
\(=x^4-2x^3+6x^2-8x+8\)
\(=x^2\left(x^2+4\right)-2x\left(x^2+4\right)+2\left(x^2+4\right)\)
\(=\left(x^2+4\right)\left(x^2-2x+2\right)\)
d) \(\left(x^2-8\right)^2+36\)
\(=x^4-16x^2+64+36\)
\(=x^4-16x^2+100\)
\(=x^4+20x^2+10^2-36x^2\)
\(=\left(x^2+10\right)^2-\left(6x\right)^2\)
\(=\left(x^2+10-6x\right)\left(x^2+10+6x\right)\)