1. **State the problem:** Simplify the expression $\left(\frac{a}{2} - 2\right) \left(\frac{a^2}{4} + a + 4\right)$.\n\n2. **Recall the distributive property:** To multiply two binomials or polynomials, multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.\n\n3. **Multiply each term:**\n$$\left(\frac{a}{2} - 2\right) \left(\frac{a^2}{4} + a + 4\right) = \frac{a}{2} \cdot \frac{a^2}{4} + \frac{a}{2} \cdot a + \frac{a}{2} \cdot 4 - 2 \cdot \frac{a^2}{4} - 2 \cdot a - 2 \cdot 4$$\n\n4. **Calculate each product:**\n$$= \frac{a^3}{8} + \frac{a^2}{2} + 2a - \frac{2a^2}{4} - 2a - 8$$\n\n5. **Simplify terms:** Note that $\frac{2a^2}{4} = \frac{a^2}{2}$. So,\n$$= \frac{a^3}{8} + \frac{a^2}{2} + 2a - \frac{a^2}{2} - 2a - 8$$\n\n6. **Cancel like terms:**\n$$= \frac{a^3}{8} + \cancel{\frac{a^2}{2}} + 2a - \cancel{\frac{a^2}{2}} - 2a - 8$$\n\n7. **Combine remaining terms:**\n$$= \frac{a^3}{8} + (2a - 2a) - 8 = \frac{a^3}{8} - 8$$\n\n**Final answer:** $$\boxed{\frac{a^3}{8} - 8}$$