2a squared _b (5a+12b)/5ab squared+12b cubed_2a squared b

To solve the given expression:

\[
2a^2 \cdot b \left(\frac{5a + 12b}{5ab^2 + 12b^3}\right) \cdot 2a^2b
\]

Let's simplify it step by step:

### Step 1: Simplify the expression inside the parentheses
\[
\frac{5a + 12b}{5ab^2 + 12b^3}
\]
This fraction can be simplified further by factoring out common terms.

Factor out \( b^2 \) from the denominator:
\[
\frac{5a + 12b}{b^2(5a + 12b)}
\]

Now, we can cancel out the common factor \( 5a + 12b \) from the numerator and the denominator:
\[
\frac{1}{b^2}
\]

### Step 2: Substitute back into the original expression
Now the expression becomes:
\[
2a^2 \cdot b \cdot \frac{1}{b^2} \cdot 2a^2b
\]

### Step 3: Simplify the product
Let's simplify this expression step by step:
- \( b \) in the numerator and one \( b^2 \) in the denominator simplify to \( \frac{1}{b} \).
- The expression now is:
  \[
  2a^2 \cdot \frac{1}{b} \cdot 2a^2b
  \]

Now, multiply the remaining terms:
\[
2a^2 \cdot 2a^2 = 4a^4
\]
\[
\frac{b}{b} = 1
\]

So, the simplified expression is:
\[
4a^4
\]

### Final Answer:
The expression simplifies to:

\[
\boxed{4a^4}
\]

Do you have any questions or would you like a more detailed explanation? 

Here are 5 related questions to consider:
1. How do you simplify fractions with polynomials in both the numerator and denominator?
2. What happens when you cancel out common terms in a fraction?
3. Can you factor out common terms from both the numerator and denominator to simplify an expression?
4. How do exponents behave when multiplying like bases?
5. What are the rules for multiplying fractions in algebra?

**Tip:** Always check for common factors in the numerator and denominator before multiplying out expressions—they can often simplify your work significantly.

Learn how to simplify the expression 2a^2 * b * (5a + 12b) / (5ab^2 + 12b^3) * 2a^2b step-by-step. This problem involves algebraic concepts such as fraction simplification and polynomial operations. Suitable for advanced high school students.

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