Simplify the following expressions fully: 1.2.1 \( \left(4+\frac{1}{x}\right)\left(2-\frac{3}{x}\right) \)

To simplify \( \left(4+\frac{1}{x}\right)\left(2-\frac{3}{x}\right) \), we can apply the distributive property (also known as the FOIL method for binomials): 1. **First**: \( 4 \cdot 2 = 8 \) 2. **Outer**: \( 4 \cdot -\frac{3}{x} = -\frac{12}{x} \) 3. **Inner**: \( \frac{1}{x} \cdot 2 = \frac{2}{x} \) 4. **Last**: \( \frac{1}{x} \cdot -\frac{3}{x} = -\frac{3}{x^2} \) Now, combine all the terms: \[ 8 - \frac{12}{x} + \frac{2}{x} - \frac{3}{x^2} \] Combine like terms (the terms with \(\frac{1}{x}\)): \[ 8 - \frac{10}{x} - \frac{3}{x^2} \] Thus, the simplified expression is: \[ 8 - \frac{10}{x} - \frac{3}{x^2} \]