1. **State the problem:** Solve the quadratic equation formed by multiplying the binomials $\{5x-2\}$ and $\{2x-3\}$. This means we want to find $x$ such that $$(5x-2)(2x-3) = 0.$$\n\n2. **Use the zero product property:** If the product of two factors is zero, then at least one of the factors must be zero. So, we set each factor equal to zero:\n$$5x - 2 = 0 \quad \text{or} \quad 2x - 3 = 0.$$\n\n3. **Solve each linear equation:**\n- For $5x - 2 = 0$:\n$$5x = 2$$\n$$x = \frac{2}{5}.$$\n- For $2x - 3 = 0$:\n$$2x = 3$$\n$$x = \frac{3}{2}.$$\n\n4. **Final answer:** The solutions to the quadratic equation are $$x = \frac{2}{5} \quad \text{and} \quad x = \frac{3}{2}.$$