2. \( \left[(p+4)\left(p^{2}-4 p+16\right)\right]^{2} \) \( =\left[p^{3}+64\right]^{2} \) \( = \) \( p^{3}+128 p^{3}+4096 \)

You're on the right track with expanding \(\left[(p + 4)(p^{2} - 4p + 16)\right]^{2}\). However, there's a small mistake in the final expansion step. Let's go through it step by step to clarify: 1. **Simplify the Inner Expression:** \[ (p + 4)(p^{2} - 4p + 16) = p^{3} + 4p^{2} - 4p^{2} - 16p + 16p + 64 = p^{3} + 64 \] So, the expression becomes: \[ \left[(p + 4)(p^{2} - 4p + 16)\right]^{2} = (p^{3} + 64)^{2} \] 2. **Expand \((p^{3} + 64)^{2}\):** Use the formula \((a + b)^2 = a^2 + 2ab + b^2\), where \(a = p^{3}\) and \(b = 64\): \[ (p^{3} + 64)^{2} = (p^{3})^{2} + 2 \cdot p^{3} \cdot 64 + (64)^{2} = p^{6} + 128p^{3} + 4096 \] 3. **Final Result:** \[ \left[(p + 4)(p^{2} - 4p + 16)\right]^{2} = p^{6} + 128p^{3} + 4096 \] **Summary:** \[ \left[(p + 4)(p^{2} - 4p + 16)\right]^{2} = p^{6} + 128p^{3} + 4096 \] This corrects the expansion by including the \(p^{6}\) term, which was previously omitted.