**Step 1 — Write the sum and product of the zeroes.**
For $px^2 + qx + r$ :
$$\alpha + \beta = -\frac{q}{p}, \qquad \alpha\beta = \frac{r}{p}$$

**Step 2 — Take $\alpha\beta$ common from the given expression.**
$$\alpha^3\beta + \beta^3\alpha = \alpha\beta\left(\alpha^2 + \beta^2\right)$$

**Step 3 — Express $\alpha^2 + \beta^2$ in terms of the sum and product.**
$$\alpha^2 + \beta^2 = (\alpha+\beta)^2 - 2\alpha\beta = \frac{q^2}{p^2} - \frac{2r}{p} = \frac{q^2 - 2pr}{p^2}$$

**Step 4 — Substitute back.**
$$\alpha^3\beta + \beta^3\alpha = \frac{r}{p} \times \frac{q^2 - 2pr}{p^2}$$

**Step 5 — Final answer.**
$$\boxed{\alpha^3\beta + \beta^3\alpha = \frac{r\left(q^2 - 2pr\right)}{p^3}}$$