FIND THE ZEROES OF 5X^3-8X^2-5X+2

5x^3 - 8x^2 - 5x  + 2 = 0 ..this doesn't appear to factor....let's use the "Rational Zeroes" Theorem to  find a root - if possible...

The possible roots  are ± 1, ± 2 and ± 2/5

I see that 2 is a root......let's use synthetic division to find the next polynomial

2       5   -8     -5     2

              10      4    -2

         5     2     -1    0

And the next polynomial is

5x^2 + 2x -1   = 0

This will not factor...using the on-site solver [and the Quadratic Formula] ...the other two roots are :

$${\mathtt{5}}{\mathtt{\,\times\,}}{{\mathtt{x}}}^{{\mathtt{2}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{2}}{\mathtt{\,\times\,}}{\mathtt{x}}{\mathtt{\,-\,}}{\mathtt{1}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\ {\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{6}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{5}}}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{0.689\: \!897\: \!948\: \!556\: \!635\: \!6}}\\ {\mathtt{x}} = {\mathtt{0.289\: \!897\: \!948\: \!556\: \!635\: \!6}}\\ \end{array} \right\}$$

Here's a graph......https://www.desmos.com/calculator/hpqotand0v