1. **State the problem:** Evaluate the expression $4 \frac{1}{2} \times \left(-\frac{2}{3}\right) + \frac{7}{8}$. 2. **Convert the mixed number to an improper fraction:** $$4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2}$$ 3. **Rewrite the expression with improper fractions:** $$\frac{9}{2} \times \left(-\frac{2}{3}\right) + \frac{7}{8}$$ 4. **Multiply the fractions:** $$\frac{9}{2} \times \left(-\frac{2}{3}\right) = \frac{9 \times (-2)}{2 \times 3} = \frac{-18}{6}$$ 5. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{18}}{\cancel{6}} = \frac{-3}{1} = -3$$ 6. **Add the remaining fraction:** $$-3 + \frac{7}{8} = \frac{-3 \times 8}{8} + \frac{7}{8} = \frac{-24}{8} + \frac{7}{8}$$ 7. **Combine the fractions:** $$\frac{-24 + 7}{8} = \frac{-17}{8}$$ 8. **Final answer:** $$-\frac{17}{8}$$ or as a mixed number: $$-2 \frac{1}{8}$$