$$ \frac { ( 3 b ) ^ { 2 } + 12 a b ^ { 5 } } { 3 b } = \square b + \square a b $$

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UpStudy AI · Accepted Answer

We start with the given expression:
$$
\frac{(3b)^2 + 12ab^5}{3b}.
$$

1. First, expand $$(3b)^2$$:
$$
(3b)^2 = 9b^2.
$$
So the numerator becomes:
$$
9b^2 + 12ab^5.
$$

2. Write the expression as two separate fractions:
$$
\frac{9b^2}{3b} + \frac{12ab^5}{3b}.
$$

3. Simplify each term individually:

- For the first term:
    $$
    \frac{9b^2}{3b} = \frac{9}{3} \cdot \frac{b^2}{b} = 3b.
    $$
    
    - For the second term:
    $$
    \frac{12ab^5}{3b} = \frac{12}{3} \cdot a \cdot \frac{b^5}{b} = 4a b^{5-1} = 4ab^4.
    $$

4. Combine the simplified terms:
$$
3b + 4ab^4.
$$

Thus, the expression simplifies to:
$$
\frac{(3b)^2 + 12ab^5}{3b} = 3b + 4ab^4.
$$

If the blanks are intended for the coefficients of each term, then they are $$3$$ in the first term and $$4$$ in the second term (noting that the second term is actually $$4ab^4$$ rather than $$4ab$$).
The expression simplifies to $$3b + 4ab^4$$.

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