To determine which option is equivalent to $5^{a + 2b}$, we can simplify this expression.

Given: $5^{a + 2b}$

Using the property of exponents that states $x^{m+n} = x^m \cdot x^n$, we can break down $5^{a + 2b}$ as: $5^{a + 2b} = 5^a \cdot 5^{2b}$

Now, let's compare this with each option:

1. Option 1: $5^a \cdot 5^2 \cdot 5^b$
   This simplifies to $5^{a + 2 + b}$, which is not equal to $5^{a + 2b}$.

2. Option 2: $5^a \cdot 25^b$
   Since $25 = 5^2$, $25^b$ is equivalent to $(5^2)^b = 5^{2b}$. Therefore, $5^a \cdot 25^b = 5^a \cdot 5^{2b} = 5^{a + 2b}$.
   This matches $5^{a + 2b}$, so Option 2 is correct.

3. Option 3: $25^{2ab}$
   This does not simplify to $5^{a + 2b}$.

4. Option 4: $25^{a + 2b}$
   This would be equivalent to $(5^2)^{a + 2b} = 5^{2(a + 2b)}$, which is not the same as $5^{a + 2b}$.

The correct answer is Option 2: $5^a \cdot 25^b$.

Would you like a more detailed breakdown or have any questions?

Here are some related questions to explore:

1. How do you expand powers with sums in the exponents?
2. Why does $25^b$ become $5^{2b}$?
3. What other exponent properties can help simplify expressions like these?
4. How would the answer change if the base were different, like $3^{a + 2b}$?
5. Can we apply these rules to non-integer exponents as well?

Tip: When you see a composite number like $25$ in an exponent problem, consider breaking it down into its prime base form (here, $5^2$) for simplification.