Sketch it, also Descartes rule of signs tells you that 1 or 3 roots are positive and none are negative.

At x=0, 10x^3-49x^2+68x-20=-20, and at x=1, 10x^3-49x^2+68x-20=9.

So there is a root between x=0 and x=1.

This narrows the candidates for the first root down to 1/2, 1/5, 2/5 and 4/5.

RonL