1. **Problem:** Solve the equation $2^{x+2} = 4\sqrt{8}$. 2. **Step 1:** Express all terms with the same base if possible. Recall that $4 = 2^2$ and $\sqrt{8} = \sqrt{2^3} = 2^{3/2}$. 3. **Step 2:** Rewrite the right side: $$4\sqrt{8} = 2^2 \times 2^{3/2} = 2^{2 + 3/2} = 2^{\frac{4}{2} + \frac{3}{2}} = 2^{\frac{7}{2}}$$ 4. **Step 3:** Now the equation is: $$2^{x+2} = 2^{\frac{7}{2}}$$ Since the bases are equal and the function $2^y$ is one-to-one, set exponents equal: $$x + 2 = \frac{7}{2}$$ 5. **Step 4:** Solve for $x$: $$x = \frac{7}{2} - 2 = \frac{7}{2} - \frac{4}{2} = \frac{3}{2}$$ 6. **Answer:** $$\boxed{x = \frac{3}{2}}$$