\(M=\frac{x^8\left(x+1\right)+x^6\left(x+1\right)+x^4\left(x+1\right)+x^2\left(x+1\right)+x+1}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{\left(x^8+x^6+x^4+x^2+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\)
\(=\frac{x^8+x^6+x^4+x^2+1}{x-1}\)
M=\(\frac{\left(x^9+x^8\right)\left(x^7+x^6\right)+...+\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\)
M=\(\frac{x^8\left(x+1\right)+x^6\left(x+1\right)+...+\left(x+1\right)}{\left(x+1\right)\left(x-1\right)}\)
M=\(\frac{\left(x+1\right)\left(x^8+x^6+x^4+x^2\right)}{\left(x+1\right)\left(x-1\right)}\)
M=\(\frac{x^8+x^6+x^4+x^2}{x-1}\)
\(M=\frac{x^8\left(x+1\right)+x^6\left(x+1\right)+x^4\left(x+1\right)+x^2\left(x+1\right)+x+1}{\left(x-1\right)\left(x+1\right)}\)
\(M=\frac{\left(x^8+x^6+x^4+x^2+x\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}\)
\(M=\frac{x^8+x^6+x^4+x^2+x}{x-1}\)
\(M=\frac{\left(x^4-x^3+x^2-x+1\right)\left(x^4+x^3+x^2+x+1\right)}{x-1}\)
