1. **State the problem:** Factor the expression $$2x^5 + 14x^3 + 20x$$. 2. **Identify the greatest common factor (GCF):** Look at the coefficients 2, 14, and 20. The GCF of these numbers is 2. Also, each term contains at least one factor of $$x$$, so the GCF for the variable part is $$x$$. 3. **Factor out the GCF:** $$2x^5 + 14x^3 + 20x = 2x(x^4) + 2x(7x^2) + 2x(10)$$ 4. **Write the factored form:** $$= 2x(x^4 + 7x^2 + 10)$$ 5. **Factor the quadratic inside the parentheses if possible:** Let $$y = x^2$$, then the expression inside is $$y^2 + 7y + 10$$. 6. **Factor the quadratic:** Find two numbers that multiply to 10 and add to 7, which are 5 and 2. 7. **Write the factorization:** $$y^2 + 7y + 10 = (y + 5)(y + 2)$$ 8. **Substitute back $$y = x^2$$:** $$= (x^2 + 5)(x^2 + 2)$$ 9. **Final factored form:** $$2x(x^2 + 5)(x^2 + 2)$$ This is the fully factored form over the real numbers.