1. **State the problem:** Simplify the expression $12(z+1)21(z+1)$. 2. **Rewrite the expression:** The expression can be seen as $12 \times (z+1) \times 21 \times (z+1)$. 3. **Group like terms:** Group the constants and the $(z+1)$ terms: $$12 \times 21 \times (z+1) \times (z+1)$$ 4. **Multiply constants:** Calculate $12 \times 21$: $$12 \times 21 = 252$$ 5. **Multiply the $(z+1)$ terms:** Since $(z+1) \times (z+1) = (z+1)^2$, rewrite as: $$252 \times (z+1)^2$$ 6. **Final simplified expression:** $$252(z+1)^2$$