1. The problem is to expand the expression $\left(x-\frac{\alpha}{\beta}\right)\left(x-\frac{\beta}{\alpha}\right)$. 2. Use the distributive property (FOIL method) to expand: $$\left(x-\frac{\alpha}{\beta}\right)\left(x-\frac{\beta}{\alpha}\right) = x \cdot x - x \cdot \frac{\beta}{\alpha} - \frac{\alpha}{\beta} \cdot x + \frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha}$$ 3. Simplify each term: - $x \cdot x = x^2$ - $-x \cdot \frac{\beta}{\alpha} = -\frac{\beta}{\alpha} x$ - $-\frac{\alpha}{\beta} \cdot x = -\frac{\alpha}{\beta} x$ - $\frac{\alpha}{\beta} \cdot \frac{\beta}{\alpha} = 1$ because the numerators and denominators cancel out. 4. Combine the terms: $$x^2 - \frac{\beta}{\alpha} x - \frac{\alpha}{\beta} x + 1$$ 5. Factor the terms with $x$: $$x^2 - \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right) x + 1$$ 6. This is the fully expanded and simplified form of the expression. Final answer: $$x^2 - \left(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\right) x + 1$$