1. **State the problem:** We need to list all positive integers $x$ such that $x^2 + 4$ is an even number less than 200. 2. **Analyze the condition:** The set $A = \{x \in \mathbb{Z}^+ : x^2 + 4 \text{ is even and } x^2 + 4 < 200\}$. 3. **Check parity:** Since 4 is even, $x^2 + 4$ is even if and only if $x^2$ is even. 4. **Recall parity rule:** $x^2$ is even if and only if $x$ is even. 5. **Inequality:** $x^2 + 4 < 200 \implies x^2 < 196 \implies x < 14$ (since $x$ is positive). 6. **List all positive even integers less than 14:** $2, 4, 6, 8, 10, 12$. 7. **Verify each:** - $2^2 + 4 = 4 + 4 = 8$ (even, < 200) - $4^2 + 4 = 16 + 4 = 20$ (even, < 200) - $6^2 + 4 = 36 + 4 = 40$ (even, < 200) - $8^2 + 4 = 64 + 4 = 68$ (even, < 200) - $10^2 + 4 = 100 + 4 = 104$ (even, < 200) - $12^2 + 4 = 144 + 4 = 148$ (even, < 200) 8. **Final answer:** $A = \{2, 4, 6, 8, 10, 12\}$