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It turns out that the "brute force" approach (i.e. asking for the roots in the (default) SR ring and building the product of the resulting monomials) gives (a not especially appetizing) radical exression :

sage: prod([(x-t[0])^t[1] for t in (x^5 + x^4 - 8*x^3 + 11*x^2 - 15*x + 2).roots()])
1/20736*(12*x + sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) + 6*sqrt(-(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) + 71/9/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) + 321/2/sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) + 43/6) + 9)*(12*x + sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) - 6*sqrt(-(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) + 71/9/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) + 321/2/sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) + 43/6) + 9)*(12*x - sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) + 6*sqrt(-(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) + 71/9/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 321/2/sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) + 43/6) + 9)*(12*x - sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) - 6*sqrt(-(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) + 71/9/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 321/2/sqrt((36*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(2/3) + 129*(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3) - 284)/(2/9*sqrt(7214)*sqrt(3) + 649/27)^(1/3)) + 43/6) + 9)*(x - 2)

viewing this monster may help you to convince yourself that is indeed the product of 5 first-degree polynomials.

But note that this is pure happenstance : your polynomial being of degree 5, its pure luck that its roots can be expressed by radicals.