1. **State the problem:** Find the Highest Common Factor (HCF) and Least Common Multiple (LCM) of the polynomials $16a^4 - 4a^2 - 4a - 1$ and $8a^3 - 1$. 2. **Factorize the first polynomial:** $$16a^4 - 4a^2 - 4a - 1$$ Group terms: $$= (16a^4 - 4a^2) - (4a + 1)$$ Factor each group: $$= 4a^2(4a^2 - 1) - 1(4a + 1)$$ Note $4a^2 - 1$ is a difference of squares: $$4a^2 - 1 = (2a - 1)(2a + 1)$$ So: $$16a^4 - 4a^2 - 4a - 1 = 4a^2(2a - 1)(2a + 1) - (4a + 1)$$ Try to factor by grouping or check if it factors further. Alternatively, try polynomial division or substitution. 3. **Factorize the second polynomial:** $$8a^3 - 1$$ This is a difference of cubes: $$8a^3 - 1 = (2a)^3 - 1^3 = (2a - 1)((2a)^2 + 2a imes 1 + 1^2) = (2a - 1)(4a^2 + 2a + 1)$$ 4. **Check for common factors:** From the second polynomial, factors are $(2a - 1)$ and $(4a^2 + 2a + 1)$. From the first polynomial, we have a factor $(2a - 1)$ inside the factorization. 5. **Divide the first polynomial by $(2a - 1)$ to find the other factor:** Perform polynomial division: $$\frac{16a^4 - 4a^2 - 4a - 1}{2a - 1} = 8a^3 + 4a^2 - 1$$ 6. **Factorize $8a^3 + 4a^2 - 1$ if possible:** Try to factor or check if it shares factors with $4a^2 + 2a + 1$. 7. **HCF:** The common factor is $(2a - 1)$. 8. **LCM:** Use the formula: $$\text{LCM} = \frac{\text{Product of polynomials}}{\text{HCF}}$$ Calculate: $$\text{LCM} = \frac{(16a^4 - 4a^2 - 4a - 1)(8a^3 - 1)}{2a - 1}$$ **Final answers:** - HCF = $2a - 1$ - LCM = $\frac{(16a^4 - 4a^2 - 4a - 1)(8a^3 - 1)}{2a - 1}$