1. **State the problem:** We are given the expression $96x^3 - 12x^2 - 112x + 14$ and a sequence of factorizations: $$2(48x^2 - 6x^2)(-56x + 7)$$ $$6x^2(8x - 1) - 7(8x + 1)$$ $$2(8x - 1)(6x^2 - 7)$$ We want to verify the factorization and understand the steps. 2. **Start with the original expression:** $$96x^3 - 12x^2 - 112x + 14$$ 3. **Group terms to factor by grouping:** Group as $(96x^3 - 12x^2) + (-112x + 14)$. 4. **Factor out common factors in each group:** From the first group: $12x^2$ is common, so $$12x^2(8x - 1)$$ From the second group: $-14$ is common, so $$-14(8x - 1)$$ 5. **Rewrite the expression:** $$12x^2(8x - 1) - 14(8x - 1)$$ 6. **Factor out the common binomial factor $(8x - 1)$:** $$ (8x - 1)(12x^2 - 14) $$ 7. **Simplify the second factor:** Factor out 2: $$ (8x - 1)(2(6x^2 - 7)) = 2(8x - 1)(6x^2 - 7) $$ 8. **Final factorization:** $$ 2(8x - 1)(6x^2 - 7) $$ **Explanation:** We used factoring by grouping, which involves grouping terms to find common factors, then factoring out the common binomial. We also factored out constants to simplify the expression. **Answer:** $$\boxed{2(8x - 1)(6x^2 - 7)}$$