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.