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Answer

$$(\frac{12}{x}-6)(x-\frac{6}{3})=\frac{-6x^2+24x-24}{x}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}(\frac{12}{x}-6)(x-\frac{6}{3})& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\left(\frac{-6x+12}{x}\right) \cdot \left(x - \frac{ 6 : \color{orangered}{ 3 } }{ 3 : \color{orangered}{ 3 }}\right) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-6x+12}{x}(x-\frac{2}{1}) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{-6x+12}{x}(x-2) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{-6x^2+24x-24}{x}\end{aligned} $$
①

Subtract $6$ from $ \dfrac{12}{x} $ to get $ \dfrac{ \color{purple}{ -6x+12 } }{ x }$.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x}$.

$$ \begin{aligned} \frac{12}{x} -6 & \xlongequal{\text{Step 1}} \frac{12}{x} - \frac{6}{\color{red}{1}} = \frac{ 12 }{ x } - \frac{ 6 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 12 } }{ x } - \frac{ \color{purple}{ 6x } }{ x }=\frac{ \color{purple}{ -6x+12 } }{ x } \end{aligned} $$
②
Divide both the top and bottom numbers by $ \color{orangered}{ 3 } $.
③

Subtract $6$ from $ \dfrac{12}{x} $ to get $ \dfrac{ \color{purple}{ -6x+12 } }{ x }$.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x}$.

$$ \begin{aligned} \frac{12}{x} -6 & \xlongequal{\text{Step 1}} \frac{12}{x} - \frac{6}{\color{red}{1}} = \frac{ 12 }{ x } - \frac{ 6 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 12 } }{ x } - \frac{ \color{purple}{ 6x } }{ x }=\frac{ \color{purple}{ -6x+12 } }{ x } \end{aligned} $$
④

Subtract $6$ from $ \dfrac{12}{x} $ to get $ \dfrac{ \color{purple}{ -6x+12 } }{ x }$.

Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{x}$.

$$ \begin{aligned} \frac{12}{x} -6 & \xlongequal{\text{Step 1}} \frac{12}{x} - \frac{6}{\color{red}{1}} = \frac{ 12 }{ x } - \frac{ 6 \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 12 } }{ x } - \frac{ \color{purple}{ 6x } }{ x }=\frac{ \color{purple}{ -6x+12 } }{ x } \end{aligned} $$
⑤
Remove 1 from denominator.
⑥

Multiply $ \dfrac{-6x+12}{x} $ by $ x-2 $ to get $ \dfrac{-6x^2+24x-24}{x} $.

Step 1: Write $ x-2 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-6x+12}{x} \cdot x-2 & \xlongequal{\text{Step 1}} \frac{-6x+12}{x} \cdot \frac{x-2}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( -6x+12 \right) \cdot \left( x-2 \right) }{ x \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -6x^2+12x+12x-24 }{ x } = \frac{-6x^2+24x-24}{x} \end{aligned} $$
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