1. **State the problem:** Find the product of $$\left(\frac{3}{5}x - \frac{y}{2}\right) \left(\frac{5}{3}x + 6y\right)$$ and verify the result for $$x = -1$$ and $$y = 2$$. 2. **Use the distributive property (FOIL method) to expand:** $$\left(\frac{3}{5}x\right)\left(\frac{5}{3}x\right) + \left(\frac{3}{5}x\right)(6y) - \left(\frac{y}{2}\right)\left(\frac{5}{3}x\right) - \left(\frac{y}{2}\right)(6y)$$ 3. **Calculate each term:** - $$\frac{3}{5}x \times \frac{5}{3}x = x^2$$ (since $$\frac{3}{5} \times \frac{5}{3} = 1$$) - $$\frac{3}{5}x \times 6y = \frac{18}{5}xy$$ - $$-\frac{y}{2} \times \frac{5}{3}x = -\frac{5}{6}xy$$ - $$-\frac{y}{2} \times 6y = -3y^2$$ 4. **Combine like terms:** $$x^2 + \left(\frac{18}{5}xy - \frac{5}{6}xy\right) - 3y^2$$ 5. **Find common denominator for the xy terms:** $$\frac{18}{5} = \frac{108}{30}, \quad \frac{5}{6} = \frac{25}{30}$$ 6. **Subtract:** $$\frac{108}{30}xy - \frac{25}{30}xy = \frac{83}{30}xy$$ 7. **Final expanded expression:** $$x^2 + \frac{83}{30}xy - 3y^2$$ 8. **Verify for $$x = -1$$ and $$y = 2$$:** $$(-1)^2 + \frac{83}{30}(-1)(2) - 3(2)^2 = 1 - \frac{166}{30} - 12 = 1 - 5.5333 - 12 = -16.5333$$ **Answer:** The product is $$x^2 + \frac{83}{30}xy - 3y^2$$ and for $$x = -1$$, $$y = 2$$, the value is approximately $$-16.5333$$.