1. **Problem:** Simplify and expand the expression $(3a + 2b)(5a - 7b)$. 2. **Formula:** Use the distributive property: $(x + y)(z + w) = xz + xw + yz + yw$. 3. **Step-by-step:** - Multiply each term in the first bracket by each term in the second bracket: $$3a \times 5a = 15a^2$$ $$3a \times (-7b) = -21ab$$ $$2b \times 5a = 10ab$$ $$2b \times (-7b) = -14b^2$$ 4. **Combine like terms:** $$15a^2 + (-21ab + 10ab) - 14b^2 = 15a^2 - 11ab - 14b^2$$ 5. **Final answer:** $$15a^2 - 11ab - 14b^2$$