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Using laplace transforms how did they solve this initial value problem of y+2y'+5y=10e^_2t to get the answer given in the problem with partial fraction expansion?

Using laplace transforms how did they solve this initial value problem of y"+2y'+5y=10e^_2t to get the answer given in the problem with partial fraction expansion? Please use ONLY partial fraction expansion I am just not sure how they got the specific answer in the photo for the IVP. Show all steps calculations scratch work and identities used so I may learn the process. Thank you

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