1. **Stating the problem:** Simplify the expression $$\frac{(4^{n+1} + 4^n)^2}{(2^{n+1} - 2^n)^2}$$. 2. **Rewrite bases:** Note that $$4 = 2^2,$$ so rewrite powers of 4 in terms of 2: $$4^{n+1} = (2^2)^{n+1} = 2^{2(n+1)} = 2^{2n+2}$$ $$4^n = (2^2)^n = 2^{2n}$$ 3. **Substitute powers back:** The numerator becomes $$(2^{2n+2} + 2^{2n})^2$$ The denominator is $$(2^{n+1} - 2^n)^2$$ 4. **Factor numerator:** Factor out the common term $$2^{2n}$$ in numerator inside the parentheses: $$2^{2n+2} + 2^{2n} = 2^{2n}(2^2 + 1) = 2^{2n}(4 + 1) = 2^{2n} \times 5$$ So numerator: $$(2^{2n} \times 5)^2 = 5^2 \times (2^{2n})^2 = 25 \times 2^{4n}$$ 5. **Factor denominator:** Factor out $$2^n$$ inside denominator: $$2^{n+1} - 2^n = 2^n (2^1 - 1) = 2^n (2 - 1) = 2^n \times 1 = 2^n$$ So denominator squared: $$(2^n)^2 = 2^{2n}$$ 6. **Combine numerator and denominator:** $$\frac{25 \times 2^{4n}}{2^{2n}} = 25 \times 2^{4n - 2n} = 25 \times 2^{2n}$$ 7. **Final answer:** $$\boxed{25 \times 2^{2n}}$$ This is the simplified form of the original expression.