Is the following (part of the) sentence correct grammatically?

... I will describe how my personal experiences and my professional and academic background have prepared me to ...

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Yes, it is correct. In general, it is fine to write A and (B and C) if B and C are grouped (both referring to background) as in your example. Another example is

Put the bananas and the green and red peppers in the fridge.

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It is correct, but because of its atypical construction, it is generally used to emphasize the conjunction of the separate elements. That is to say, the extra stop with an added conjunction emphasizes both the nature of the conjunction and helps restate the independence of each component part.

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