This question tests your understanding of converting direct speech into indirect speech, specifically focusing on changes in modal verbs and demonstratives. When transforming "He said, 'I can solve this problem'" into indirect speech, several key rules apply.
Firstly, the reporting verb "said" indicates that the sentence is in the past tense, requiring a backshift in the tense of the reported clause. The modal verb "can" in direct speech changes to "could" in indirect speech to reflect this past context. This is a standard rule for modal verbs like "will" becoming "would," "may" becoming "might," and "shall" becoming "should."
Secondly, the pronoun "I" changes to "he" to match the subject of the reporting verb. Thirdly, the demonstrative pronoun "this" typically changes to "that" or "the" in indirect speech, depending on context, to indicate a shift in proximity or time. In this case, "this problem" becomes "the problem."
Therefore, combining these changes, the correct indirect speech form is "He said that he could solve the problem." Options A, C, and D fail to apply the correct tense backshift or pronoun changes, making them incorrect.