c) \( \frac{3}{4}\left(\frac{1}{3} a-b\right)\left(\frac{9}{3} a+b\right) \)

To simplify the expression \( \frac{3}{4}\left(\frac{1}{3} a-b\right)\left(\frac{9}{3} a+b\right) \), we can start by rewriting the factors: 1. The term \(\frac{9}{3}\) simplifies to \(3\). So the expression becomes: \[ \frac{3}{4}\left(\frac{1}{3}a - b\right)(3a + b) \] 2. Next, we can distribute the factors. Using the distributive property (also known as the FOIL method for binomials), we get: \[ \left(\frac{1}{3} a - b\right)(3a + b) = \frac{1}{3} a \cdot 3a + \frac{1}{3} a \cdot b - b \cdot 3a - b \cdot b \] 3. Now let's simplify each part: - \( \frac{1}{3} a \cdot 3a = a^2 \) - \( \frac{1}{3} a \cdot b = \frac{1}{3} ab \) - \( -b \cdot 3a = -3ab \) - \( -b \cdot b = -b^2 \) 4. Combine like terms: \[ a^2 + \frac{1}{3} ab - 3ab - b^2 = a^2 - \frac{8}{3} ab - b^2 \] 5. Finally, we multiply the entire expression by \(\frac{3}{4}\): \[ \frac{3}{4}\left(a^2 - \frac{8}{3} ab - b^2\right) = \frac{3}{4}a^2 - 2ab - \frac{3}{4}b^2 \] Thus, the simplified expression is: \[ \frac{3}{4}a^2 - 2ab - \frac{3}{4}b^2 \]