'Project' is a singular noun, so it will be followed by a singular verb, so the use of 'approaches' is correct. All prepositions are followed by either a noun or a gerund, so the use of 'examining' will be correct here.
The correct sentence is- 'The project approaches this question by examining the financial system in a country which has done well economically.'
Hence, option (d) is correct.
NOTE/-
Gerund: Gerund is a verb form that functions as a noun, ending in -ing. A gerund can be either the subject or object of a sentence in English. Example: "Everybody enjoyed her dancing and singing at the party." The words 'dancing' and 'singing' are gerunds.
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