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\left(4x^{2}+9\right)\left(x^{2}+4\right)
Find one factor of the form kx^{m}+n, where kx^{m} divides the monomial with the highest power 4x^{4} and n divides the constant factor 36. One such factor is 4x^{2}+9. 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}+4,4x^{2}+9.