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The expression you have provided is:

\[
3a^2 \left( 1 + 3ab^2 \right) + a \left( a^2 + 4b \right)
\]

Let's simplify this step by step:

### Step 1: Distribute \( 3a^2 \) across the first bracket
\[
3a^2(1 + 3ab^2) = 3a^2 \cdot 1 + 3a^2 \cdot 3ab^2 = 3a^2 + 9a^3b^2
\]

### Step 2: Distribute \( a \) across the second bracket
\[
a(a^2 + 4b) = a \cdot a^2 + a \cdot 4b = a^3 + 4ab
\]

### Step 3: Combine all terms
Now, put everything together:
\[
3a^2 + 9a^3b^2 + a^3 + 4ab
\]

### Step 4: Combine like terms (if any)
There are two \( a^3 \) terms:
\[
9a^3b^2 + a^3 + 3a^2 + 4ab
\]

This is the simplified expression.

Would you like further details or have any questions about the process?

### Related Questions:
1. How would this expression change if \( b = 1 \)?
2. Can we factor out common terms in this expression?
3. How does the term \( 9a^3b^2 \) affect the overall degree of the polynomial?
4. What are the degrees of each term in this expression?
5. How would you approach simplifying similar expressions involving exponents?

### Tip:
When simplifying expressions, always distribute terms carefully and look for like terms to combine.

Simplify the expression 3a^2(1 + 3ab^2) + a(a^2 + 4b).

Learn how to simplify the polynomial expression 3a^2(1 + 3ab^2) + a(a^2 + 4b) with a step-by-step breakdown. This problem covers key algebraic concepts, including distribution and combining like terms, suitable for students in Grades 8-10.

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