1. The problem is to multiply the numbers $2$, $-\frac{2}{3}$, and $\frac{3}{4}$.\n\n2. The multiplication rule for fractions and integers is to multiply the numerators together and the denominators together.\n\n3. Write the expression: $$2 \times \left(-\frac{2}{3}\right) \times \frac{3}{4}$$\n\n4. Convert the integer 2 to a fraction: $$\frac{2}{1} \times \left(-\frac{2}{3}\right) \times \frac{3}{4}$$\n\n5. Multiply the numerators: $$2 \times (-2) \times 3 = -12$$\n\n6. Multiply the denominators: $$1 \times 3 \times 4 = 12$$\n\n7. So the product is: $$\frac{-12}{12}$$\n\n8. Simplify the fraction by canceling common factors: $$\frac{\cancel{12} \times (-1)}{\cancel{12}} = -1$$\n\n9. Therefore, the final answer is $-1$.