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\left(x^{2}-8\right)\left(x^{2}+1\right)
Find one factor of the form x^{k}+m, where x^{k} divides the monomial with the highest power x^{4} and m divides the constant factor -8. One such factor is x^{2}-8. Factor the polynomial by dividing it by this factor. The following polynomials are not factored since they do not have any rational roots: x^{2}-8,x^{2}+1.