$${x^8} - {x^7} - {x^6} + {x^5} + 3{x^4} - 4{x^3} - 2{x^2} + 4x - 1 = 0$$

$$ \Rightarrow {x^7}(x - 1) - {x^5}(x - 1) + 3{x^3}(x - 1) - x({x^2} - 1) + 2x(1 - x) + (x - 1) = 0$$

$$ \Rightarrow (x - 1)({x^7} - {x^5} + 3{x^3} - x(x + 1) - 2x + 1) = 0$$

$$ \Rightarrow (x - 1)({x^7} - {x^5} + 3{x^3} - {x^2} - 3x + 1) = 0$$

$$ \Rightarrow (x - 1)({x^5}({x^2} - 1) + 3x({x^2} - 1) - 1({x^2} - 1)) = 0$$

$$ \Rightarrow (x - 1)({x^2} - 1)({x^5} + 3x - 1) = 0$$

$$\therefore$$ $$x = \, \pm \,1$$ are roots of above equation and $${x^5} + 3x - 1$$ is a monotonic term hence vanishs at exactly one value of x other than 1 or $$-$$1.

$$\therefore$$ 3 real roots.