1. **State the problem:** Find the roots of the polynomial function $$f(x) = 4x^5 - 8x^4 - 5x^3 + 10x^2 + x - 2.$$\n\n2. **Recall the formula and rules:** To find roots, solve $$f(x) = 0.$$ We can try factoring by grouping or use rational root theorem to test possible roots.\n\n3. **Apply Rational Root Theorem:** Possible rational roots are factors of constant term $-2$ over factors of leading coefficient $4$, i.e., $$\pm 1, \pm 2, \pm \frac{1}{2}, \pm \frac{1}{4}.$$\n\n4. **Test $x=1$: $$f(1) = 4 - 8 - 5 + 10 + 1 - 2 = 0.$$ So, $x=1$ is a root.\n\n5. **Divide $f(x)$ by $(x-1)$ using synthetic division:**\n$$\begin{array}{r|rrrrrr} 1 & 4 & -8 & -5 & 10 & 1 & -2 \\ \hline & 4 & -4 & -9 & 1 & 2 & 0 \end{array}$$\nThe quotient is $$4x^4 - 4x^3 - 9x^2 + x + 2.$$\n\n6. **Factor the quotient:** Try to factor $$4x^4 - 4x^3 - 9x^2 + x + 2.$$\n\n7. **Try factoring as product of quadratics:** $$(ax^2 + bx + c)(dx^2 + ex + f) = 4x^4 - 4x^3 - 9x^2 + x + 2.$$\n\n8. **Find coefficients:** After trial, factorization is $$(4x^2 + 3x - 2)(x^2 - x - 1).$$\n\n9. **Solve each quadratic:**\n- For $$4x^2 + 3x - 2 = 0,$$ use quadratic formula: $$x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 4 \cdot (-2)}}{2 \cdot 4} = \frac{-3 \pm \sqrt{9 + 32}}{8} = \frac{-3 \pm \sqrt{41}}{8}.$$\n\n- For $$x^2 - x - 1 = 0,$$ use quadratic formula: $$x = \frac{1 \pm \sqrt{1 + 4}}{2} = \frac{1 \pm \sqrt{5}}{2}.$$\n\n10. **List all roots:**\n$$x = 1, \quad x = \frac{-3 + \sqrt{41}}{8}, \quad x = \frac{-3 - \sqrt{41}}{8}, \quad x = \frac{1 + \sqrt{5}}{2}, \quad x = \frac{1 - \sqrt{5}}{2}.$$