We are given the following functions:
[tex]\begin{gathered} P(x)=\frac{2}{3x-1} \\ \\ Q(x)=\frac{6}{-3x+2} \end{gathered}[/tex]
We are asked to determine:
[tex]P(x)\div Q(x)[/tex]
This is equivalent to:
[tex]\frac{P(x)}{Q(x)}[/tex]
Substituting the functions:
[tex]\frac{P(x)}{Q(x)}=\frac{\frac{2}{3x-1}}{\frac{6}{-3x+2}}[/tex]
Now, we use the following property:
[tex]\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}*\frac{d}{c}[/tex]
Applying the property we get:
[tex]\frac{P(x)}{Q(x)}=(\frac{2}{3x-1})(\frac{-3x+2}{6})[/tex]
Solving the products:
[tex]\frac{P(x)}{Q(x)}=\frac{2(-3x+2)}{6(3x-1)}[/tex]
Simplifying we get:
[tex]\frac{P(x)}{Q(x)}=\frac{-3x+2}{3(3x-1)}[/tex]
And thus we get the desired expression.
Now, we are asked to determine:
[tex]P(x)*Q(x)[/tex]
This is the product of the functions. Substituting we get:
[tex]P(x)*Q(x)=(\frac{2}{3x-1})(\frac{6}{-3x+2})[/tex]
Solving the products:
[tex]P(x)*Q(x)=\frac{12}{(3x-1)(-3x+2)}[/tex]
Since we can't simplify any further this is the final answer.
