Step 1: Understanding the preposition use.
The phrase “Due to” is used to indicate the cause or reason for something.
Step 2: Explanation.
Here, the sentence refers to the car being ruined because of his carelessness. Thus, “Due to his carelessness” correctly completes the sentence.
Step 3: Conclusion.
Hence, the correct sentence is: “Due to his carelessness, our car was ruined.”