Step 1 :First, we calculate the logarithm base 6 of 4, which is approximately 0.7737056144690831.
Step 2 :Next, we calculate the logarithm base 6 of 3 squared, which is approximately 1.226294385530917.
Step 3 :We then add the two results together, which gives us a final result of 2.0.
Step 4 :Therefore, the expression \( \log _{6} 4+2 \log _{6} 3 \) simplifies to 2.
Step 5 :Final Answer: \(\boxed{2}\)