1. The problem is to simplify the expression $2^4 \times 2^{n+2} - 2 \times 2^{n+2}$ and check if it can be written as $2^4 \times -2 + 2^{n+2} \times 2^{n+2}$. 2. Recall the laws of exponents: when multiplying powers with the same base, add the exponents: $$a^m \times a^n = a^{m+n}$$ 3. Simplify the first term: $$2^4 \times 2^{n+2} = 2^{4 + (n+2)} = 2^{n+6}$$ 4. The second term is: $$2 \times 2^{n+2} = 2^1 \times 2^{n+2} = 2^{1 + n + 2} = 2^{n+3}$$ 5. So the expression becomes: $$2^{n+6} - 2^{n+3}$$ 6. Factor out the common term $2^{n+3}$: $$2^{n+3} \times (2^3 - 1) = 2^{n+3} \times (8 - 1) = 2^{n+3} \times 7$$ 7. Therefore, the simplified form is: $$7 \times 2^{n+3}$$ 8. The proposed rewriting $2^4 \times -2 + 2^{n+2} \times 2^{n+2}$ is incorrect because it changes the structure and values of the original expression. Final answer: $$7 \times 2^{n+3}$$