$$ \frac { ( 3 b ) ^ { 2 } + 12 a b ^ { 5 } } { 3 b } = 6 b + \square a b $$

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UpStudy AI · Accepted Answer

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

**Step 1. Expand the numerator**

Calculate $$(3b)^2$$:
$$
(3b)^2=9b^2.
$$
So the numerator becomes
$$
9b^2+12ab^5.
$$

**Step 2. Divide each term by $$3b$$**

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

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

Thus, the simplified result is
$$
3b+4ab^4.
$$

**Step 3. Write $$4ab^4$$ in the form “$$\square\, ab$$”**

We wish to express the term $$4ab^4$$ in the form
$$
\square\,ab.
$$
Notice that
$$
4ab^4 = 4b^3\cdot ab,
$$
since
$$
ab\cdot b^3 = ab^4.
$$

Therefore, we have
$$
\frac{(3b)^2+12ab^5}{3b} = 3b+4ab^4 = 3b+ (4b^3)ab.
$$

Comparing with the form 
$$
6b+\square\, ab,
$$
we see that if we want consistency the term multiplying $$ab$$ must carry the factor $$4b^3$$. (One may note that the constant term here is $$3b$$ rather than $$6b$$; thus, if the textbook expression is intended to have $$6b$$ as the first term, then a correction might be needed. But as given, the simplification shows the missing factor in the $$ab$$ term is $$4b^3$$.)

**Final Answer:**
$$
\square = 4b
\[ \square = 4b^3 $$

\( \frac { ( 3 b ) ^ { 2 } + 12 a b ^ { 5 } } { 3 b } = 6 b + \square a b \)

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