1. **State the problem:** Simplify the expression $\left(\frac{4}{5} - \frac{1}{12}\right) \times \frac{2}{8}$.\n\n2. **Find a common denominator to subtract the fractions inside the parentheses:** The denominators are 5 and 12. The least common denominator (LCD) is 60.\n\n3. **Convert each fraction:**\n$$\frac{4}{5} = \frac{4 \times 12}{5 \times 12} = \frac{48}{60}$$\n$$\frac{1}{12} = \frac{1 \times 5}{12 \times 5} = \frac{5}{60}$$\n\n4. **Subtract the fractions:**\n$$\frac{48}{60} - \frac{5}{60} = \frac{48 - 5}{60} = \frac{43}{60}$$\n\n5. **Multiply the result by $\frac{2}{8}$:**\n$$\frac{43}{60} \times \frac{2}{8} = \frac{43 \times 2}{60 \times 8} = \frac{86}{480}$$\n\n6. **Simplify the fraction $\frac{86}{480}$:**\nBoth numerator and denominator are divisible by 2:\n$$\frac{\cancel{86}^{43}}{\cancel{480}^{240}} = \frac{43}{240}$$\n\n7. **Final answer:**\n$$\boxed{\frac{43}{240}}$$