1. We are asked to expand and simplify the expression **$(3a + 4b)(3a - 1b)$**. 2. Use the distributive property (FOIL method) to multiply each term: $$ (3a + 4b)(3a - 1b) = 3a \times 3a + 3a \times (-1b) + 4b \times 3a + 4b \times (-1b) $$ 3. Calculate each term: $$ 3a \times 3a = 9a^2 $$ $$ 3a \times (-1b) = -3ab $$ $$ 4b \times 3a = 12ab $$ $$ 4b \times (-1b) = -4b^2 $$ 4. Combine like terms: $$ 9a^2 + (-3ab) + 12ab + (-4b^2) = 9a^2 + 9ab - 4b^2 $$ 5. The simplified product is: $$\boxed{9a^2 + 9ab - 4b^2}$$