1. **State the problem:** Factor the polynomial $$14x^{2n + 1} + 7x^{n + 3} - 21x^{n + 2}$$ completely. 2. **Identify common factors:** Look for the greatest common factor (GCF) in all terms. - Coefficients: GCF of 14, 7, and 21 is 7. - Variable powers: The smallest power of $x$ among the terms is $x^n$ (since powers are $2n+1$, $n+3$, and $n+2$). 3. **Factor out the GCF:** $$7x^n$$ 4. **Divide each term by the GCF:** - $$\frac{14x^{2n+1}}{7x^n} = 2x^{(2n+1)-n} = 2x^{n+1}$$ - $$\frac{7x^{n+3}}{7x^n} = x^{(n+3)-n} = x^3$$ - $$\frac{-21x^{n+2}}{7x^n} = -3x^{(n+2)-n} = -3x^2$$ 5. **Write the factored form:** $$7x^n(2x^{n+1} + x^3 - 3x^2)$$ **Final answer:** $$7x^n(2x^{n+1} + x^3 - 3x^2)$$