Let's carefully match each expression with its simplified form, considering the functions [tex]\( P(x) = \frac{2}{3x - 1} \)[/tex] and [tex]\( Q(x) = \frac{6}{-3x + 2} \)[/tex].
Given:
[tex]\[
P(x) \cdot Q(x)
\][/tex]
To multiply [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex]:
[tex]\[
P(x) \cdot Q(x) = \frac{2}{3x - 1} \cdot \frac{6}{-3x + 2}
\][/tex]
We know that the simplified form of [tex]\( P(x) \cdot Q(x) \)[/tex] is:
[tex]\[
\frac{12}{(3x - 1)(-3x + 2)}
\][/tex]
Next, given:
[tex]\[
P(x) \div Q(x)
\][/tex]
To divide [tex]\( P(x) \)[/tex] by [tex]\( Q(x) \)[/tex]:
[tex]\[
P(x) \div Q(x) = \frac{2}{3x - 1} \div \frac{6}{-3x + 2} = \frac{2}{3x - 1} \cdot \frac{-3x + 2}{6}
\][/tex]
We know that the simplified form of [tex]\( P(x) \div Q(x) \)[/tex] is:
[tex]\[
2 \left(\frac{1}{3} - \frac{x}{2}\right) / (3x - 1) = \frac{-2(12x - 5)}{(3x-1)(-3x+2)}
\][/tex]
Now, we can match them with their simplified expressions:
[tex]\[
\begin{array}{l}
P(x) \cdot Q(x) \\
\xrightarrow{\text{ }} \\
\frac{12}{(3x-1)(-3x+2)}\\
\\
P(x) \div Q(x) \\
\xrightarrow{\text{ }} \\
\frac{-2(12x-5)}{(3x-1)(-3x+2)} \\
\end{array}
\][/tex]
