1. **State the problem:** Factor the polynomial $$5x^5 - 20x^4 + 5x^3 + 50x^2 - 20x - 40$$. 2. **Factor out the greatest common factor (GCF):** Each term is divisible by 5, so factor out 5: $$5(x^5 - 4x^4 + x^3 + 10x^2 - 4x - 8)$$ 3. **Group terms to factor by grouping:** Group the terms as follows: $$(x^5 - 4x^4 + x^3) + (10x^2 - 4x - 8)$$ 4. **Factor each group:** - From the first group, factor out $$x^3$$: $$x^3(x^2 - 4x + 1)$$ - From the second group, factor out 2: $$2(5x^2 - 2x - 4)$$ 5. **Rewrite the expression:** $$5[x^3(x^2 - 4x + 1) + 2(5x^2 - 2x - 4)]$$ 6. **Check if further factoring or common factors exist:** The two terms inside the bracket do not share a common binomial factor, so try to factor the original polynomial differently. 7. **Try factoring the original polynomial by grouping differently:** Group as: $$(5x^5 - 20x^4 + 5x^3) + (50x^2 - 20x - 40)$$ 8. **Factor each group:** - From the first group, factor out $$5x^3$$: $$5x^3(x^2 - 4x + 1)$$ - From the second group, factor out 10: $$10(5x^2 - 2x - 4)$$ 9. **Rewrite:** $$5x^3(x^2 - 4x + 1) + 10(5x^2 - 2x - 4)$$ 10. **Try to factor the quadratic expressions:** - Factor $$x^2 - 4x + 1$$ using the quadratic formula: $$x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3}$$ So it factors as: $$(x - (2 + \sqrt{3}))(x - (2 - \sqrt{3}))$$ - Factor $$5x^2 - 2x - 4$$: Find two numbers that multiply to $$5 \times (-4) = -20$$ and add to $$-2$$: these are 2 and -10. Rewrite: $$5x^2 - 10x + 8x - 4 = 5x(x - 2) + 2(4x - 2)$$ This does not factor nicely, so use quadratic formula: $$x = \frac{2 \pm \sqrt{4 + 80}}{10} = \frac{2 \pm \sqrt{84}}{10} = \frac{2 \pm 2\sqrt{21}}{10} = \frac{1 \pm \sqrt{21}}{5}$$ 11. **Final factored form:** $$5[x^3(x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) + 2(5x^2 - 2x - 4)]$$ Since the second quadratic does not factor nicely, the polynomial is factored as much as possible over the reals with radicals. **Answer:** $$5x^3(x - (2 + \sqrt{3}))(x - (2 - \sqrt{3})) + 10(5x^2 - 2x - 4)$$