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\left(7x-1\right)\left(-49x^{2}-7x-1\right)
By Rational Root Theorem, all rational roots of a polynomial are in the form \frac{p}{q}, where p divides the constant term 1 and q divides the leading coefficient -343. One such root is \frac{1}{7}. Factor the polynomial by dividing it by 7x-1. Polynomial -49x^{2}-7x-1 is not factored since it does not have any rational roots.