1. We are asked to multiply $3.4 \times 10^2$ by $4 \times 10^4$. 2. Multiply the decimal parts: $3.4 \times 4 = 13.6$. 3. Multiply the powers of ten by adding their exponents: $10^2 \times 10^4 = 10^{2+4} = 10^6$. 4. Combine these results: $13.6 \times 10^6$. 5. Compare with the given options: A. $3.8 \times 10^8$, B. $13.6 \times 10^8$, C. $13.6 \times 10^6$, D. $3.8 \times 10^6$. 6. The correct answer is option C: $13.6 \times 10^6$.