The number of distinct real roots of the equation $\left({\mathrm{x}}^3-{\mathrm{x}}^2-\mathrm{x}+1\right){\mathrm{x}}^5+\mathrm{x}\left(3{\mathrm{x}}^3-4{\mathrm{x}}^2-2\mathrm{x}+4\right)-1=0$ is _________

$\begin{array}{r}{x}^5\left(x^3-x^2-x+1\right)+x\left(3x^3-4x^2-2x+4\right)-1=0\\ x^8-x^7-x^6+x^5+3x^4-4x^3-2x^2+4x-1=0\\ \begin{array}{r}\Rightarrow (x-1)\left(x^7-x^5+3x^3-x^2-3x+1\right)=0\\ \Rightarrow (x-1)\left(x^5\left(x^2-1\right)+3x\left(x^2-1\right)-1\left(x^2-1\right)\right)=0\\ \Rightarrow (x-1)\left(x^2-1\right)\left(x^5+3x-1\right)=0\end{array}\\ \Rightarrow (x-1{)}^2(x+1)\left(x^5+3x-1\right)=0\\ \text{ Let }f\left(x\right)=x^5+3x-1\\ \mathrm{f}\text{'}\left(\mathrm{x}\right)>0\mathrm{\forall }\mathrm{x}\in \mathrm{R}\\ x^5+3x-1\text{ is a monotonic function }\\ \mathrm{hence} \mathrm{vanish}es \mathrm{at} \mathrm{exactly} \mathrm{one} \mathrm{value} \mathrm{of} x other than 1,-1\\ \text{ Hence }3\text{ real distinct roots. }\end{array}$