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\left(3x^{2}+1\right)\left(x^{2}-7\right)
Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 3x^{4} and n divides the constant factor -7. One such factor is 3x^{2}+1. 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}-7,3x^{2}+1.