1. **State the problem:** Find the values of $A$, $B$, and $C$ in the expansion of $(4x - 3y)(x + 2y) = Ax^2 + Bxy + Cy^2$. 2. **Use the distributive property (FOIL) to expand:** $$(4x - 3y)(x + 2y) = 4x \cdot x + 4x \cdot 2y - 3y \cdot x - 3y \cdot 2y$$ 3. **Calculate each term:** $$4x \cdot x = 4x^2$$ $$4x \cdot 2y = 8xy$$ $$-3y \cdot x = -3xy$$ $$-3y \cdot 2y = -6y^2$$ 4. **Combine like terms:** $$4x^2 + (8xy - 3xy) - 6y^2 = 4x^2 + 5xy - 6y^2$$ 5. **Identify coefficients:** $$A = 4, \quad B = 5, \quad C = -6$$ **Final answer:** $$A=4, \quad B=5, \quad C=-6$$