Eigenvalue and eigenvector problem

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I am looking to get to the answer of Question 3 e). I successfully expand the determinant to -λ^3 + 6λ^2 + 3λ - 13 using the characteristic equation and this is where I get stuck. I tried rational root theorem, synthetic division, factoring, yet I still can't get the roots and hence the answer in the end. Any solutions or pointers are appreciated.