1. **State the problem:** Simplify the expression $$5 \cdot (25)^{n+1} - 25 \cdot 5^{2n}$$. 2. **Recall the rules:** - $25 = 5^2$. - When multiplying powers with the same base, add exponents: $a^m \cdot a^n = a^{m+n}$. 3. **Rewrite terms with base 5:** $$5 \cdot (25)^{n+1} = 5 \cdot (5^2)^{n+1} = 5 \cdot 5^{2(n+1)} = 5 \cdot 5^{2n+2}$$ 4. **Simplify the first term:** $$5 \cdot 5^{2n+2} = 5^{1} \cdot 5^{2n+2} = 5^{1 + 2n + 2} = 5^{2n + 3}$$ 5. **Rewrite the second term:** $$25 \cdot 5^{2n} = 5^2 \cdot 5^{2n} = 5^{2 + 2n} = 5^{2n + 2}$$ 6. **Substitute back:** $$5^{2n + 3} - 5^{2n + 2}$$ 7. **Factor out the common term $5^{2n + 2}$:** $$5^{2n + 2} \left(5^{1} - 1\right) = 5^{2n + 2} (5 - 1) = 5^{2n + 2} \cdot 4$$ **Final answer:** $$\boxed{4 \cdot 5^{2n + 2}}$$