1. **Stating the problem:** We need to factor common terms from the expression $$y'=(4x+3)^3(x+1) - 4\left[16(x+1) - 3(4x+3)\right].$$ 2. **Simplify the expression inside the brackets:** Calculate inside the bracket: $$16(x+1) - 3(4x+3) = 16x + 16 - 12x - 9 = (16x - 12x) + (16 - 9) = 4x + 7.$$ Now rewrite the expression: $$y' = (4x+3)^3(x+1) - 4(4x+7).$$ 3. **Check for common factors:** The expression is: $$y' = (4x+3)^3(x+1) - 4(4x+7).$$ Look for any common binomial or factor but note that $(4x+7)$ and $(4x+3)(x+1)$ are different. 4. **Try to factor by grouping or as difference if possible:** Since no obvious common factor exists between the two terms, the factorization focuses on simplifying the second term or looking for structure but no common factor exists to factor out here fully. 5. **Hence the simplified form with expression inside bracket simplified is:** $$y' = (4x+3)^3(x+1) - 4(4x + 7).$$ This is as factored as possible given the different terms. **Final answer:** The simplified expression factoring the bracket is $$y' = (4x+3)^3(x+1) - 4(4x+7).$$