Sure! Let's work through the given expression step by step and expand it thoroughly. The expression we need to expand is:
[tex]\[
(5a^3b + 3a^2b^2 - 7ab^3 - 12b^5) \cdot (6a^{-3b}) \cdot (4a^2 + 2b)
\][/tex]
Step 1: Multiply the first two parts of the expression
First, let's distribute the second term [tex]\(6a^{-3b}\)[/tex] through the first term:
[tex]\[
(5a^3b \cdot 6a^{-3b}) + (3a^2b^2 \cdot 6a^{-3b}) + (-7ab^3 \cdot 6a^{-3b}) + (-12b^5 \cdot 6a^{-3b})
\][/tex]
This results in:
[tex]\[
30a^{3-3b}b + 18a^{2-3b}b^2 - 42a^{1-3b}b^3 - 72a^{-3b}b^5
\][/tex]
So, rewriting the intermediate expression:
[tex]\[
30a^{3-3b}b + 18a^{2-3b}b^2 - 42a^{1-3b}b^3 - 72a^{-3b}b^5
\][/tex]
Step 2: Multiply this result by the third part [tex]\((4a^2 + 2b)\)[/tex]
Now, we need to distribute [tex]\((4a^2 + 2b)\)[/tex] through the intermediate expression:
[tex]\[
\begin{aligned}
&\left(30a^{3-3b}b\right)(4a^2) + \left(30a^{3-3b}b\right)(2b) + \left(18a^{2-3b}b^2\right)(4a^2) + \left(18a^{2-3b}b^2\right)(2b) \\
&+ \left(-42a^{1-3b}b^3\right)(4a^2) + \left(-42a^{1-3b}b^3\right)(2b) + \left(-72a^{-3b}b^5\right)(4a^2) + \left(-72a^{-3b}b^5\right)(2b)
\end{aligned}
\][/tex]
Simplifying each term, we get:
[tex]\[
\begin{aligned}
& 120a^{5-3b}b + 60a^{3-3b}b^2 + 72a^{4-3b}b^2 + 36a^{2-3b}b^3 \\
& -168a^{3-3b}b^3 - 84a^{1-3b}b^4 - 288a^{2-3b}b^5 - 144a^{-3b}b^6
\end{aligned}
\][/tex]
Step 3: Simplify the expression by combining like terms
There are no like terms in this expansion, so the expanded and simplified expression is:
[tex]\[
120a^{5-3b}b + 60a^{3-3b}b^2 + 72a^{4-3b}b^2 + 36a^{2-3b}b^3 - 168a^{3-3b}b^3 - 84a^{1-3b}b^4 - 288a^{2-3b}b^5 - 144a^{-3b}b^6
\][/tex]
Thus, the fully expanded solution to the given expression is:
[tex]\[
120\frac{a^5b}{a^{3b}} + 72\frac{a^4b^2}{a^{3b}} - 168\frac{a^3b^3}{a^{3b}} + 60\frac{a^3b^2}{a^{3b}} - 288\frac{a^2b^5}{a^{3b}} + 36\frac{a^2b^3}{a^{3b}} - 84\frac{a b^4}{a^{3b}} - 144\frac{b^6}{a^{3b}}
\][/tex]
