\(=x^{10}\left(x+1\right)+x^8\left(x+1\right)+...+x^4\left(x+1\right)+x^2\left(x+1\right)+\left(x+1\right)\)
\(=\left(x+1\right)\left(x^{10}+x^8+x^6+x^4+x^2+1\right)\)
\(=\left(x+1\right)\left(x^2+1\right)\left(x^8+x^4+1\right)\)
\(=\left(x+1\right)\left(x^2+1\right)\left[x^8+2x^4+1-x^4\right]\)
\(=\left(x+1\right)\left(x^2+1\right)\left(x^4+1-x^2\right)\left(x^4+1+x^2\right)\)
\(=\left(x+1\right)\left(x^2+1\right)\left(x^4-x^2+1\right)\left(x^4+2x^2+1-x^2\right)\)
\(=\left(x+1\right)\left(x^2+1\right)\left(x^4-x^2+1\right)\left(x^2+1-x\right)\left(x^2+1+x\right)\)
x11 + x10 + x9 + x8 + x7 + x6 + x5 + x4 + x3 + x2 + x + 1
= x10( x + 1) + x8( x + 1) + x6( x + 1) + x4( x + 1) + x2( x + 1) + ( x + 1)
= (x+1).(x10 + x8 + x6 + x4 + x2 + 1)
=(x+1).[x8.(x2 + 1) + x4.(x + 1) + (x2 + 1)]
=(x+1).(x2 + 1).(x8 + x4 + 1)
