1. The problem is to analyze the polynomial $$x^4 + 20x^3 - 7x^2 - 80x + 12$$ including its roots, factorization, and general properties. 2. First, try to find rational roots using the Rational Root Theorem. Possible rational roots are factors of 12 divided by factors of 1, i.e., $$\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$$. 3. Test $$x = 1$$: $$1 + 20 - 7 - 80 + 12 = -54 \ne 0$$. Test $$x = 2$$: $$16 + 160 - 28 - 160 + 12 = 0$$, so $$x=2$$ is a root. 4. Divide the polynomial by $$x-2$$ using synthetic or polynomial division: $$x^4 + 20x^3 - 7x^2 - 80x + 12 \div (x-2) = x^3 + 22x^2 + 37x - 6$$. 5. Repeat root testing on $$x^3 + 22x^2 + 37x - 6$$. Try $$x=1$$: $$1 + 22 + 37 - 6 = 54 \ne 0$$. 6. Try $$x = -1$$: $$-1 + 22 - 37 - 6 = -22 \ne 0$$. Try $$x= -2$$: $$-8 + 88 - 74 - 6 = 0$$, so $$x=-2$$ is a root. 7. Divide $$x^3 + 22x^2 + 37x - 6$$ by $$x+2$$: $$x^2 + 20x + 3$$. 8. Factor the quadratic $$x^2 + 20x + 3$$ using the quadratic formula: $$x = \frac{-20 \pm \sqrt{20^2 - 4 \cdot 1 \cdot 3}}{2} = \frac{-20 \pm \sqrt{400 - 12}}{2} = \frac{-20 \pm \sqrt{388}}{2} = \frac{-20 \pm 2\sqrt{97}}{2} = -10 \pm \sqrt{97}$$. 9. Thus, the factorization is: $$ (x - 2)(x + 2)(x^2 + 20x + 3) $$ or fully factored as $$ (x-2)(x+2)(x - (-10 + \sqrt{97}))(x - (-10 - \sqrt{97})) $$. 10. The roots are $$2, -2, -10 + \sqrt{97}, -10 - \sqrt{97}$$. 11. Summary: We found all roots and factorized the polynomial completely.