$$x(x-9)\dfrac{x+4}{6}x^3(x+4)(x-9)=\dfrac{x^8-10x^7-47x^6+360x^5+1296x^4}{6}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}x(x-9)\frac{x+4}{6}x^3(x+4)(x-9)& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}(x^2-9x)\frac{x+4}{6}x^3(x+4)(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{x^3-5x^2-36x}{6}x^3(x+4)(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{x^6-5x^5-36x^4}{6}(x+4)(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{x^7-x^6-56x^5-144x^4}{6}(x-9) \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{x^8-10x^7-47x^6+360x^5+1296x^4}{6}\end{aligned} $$ | |
| ① | Multiply $ \color{blue}{x} $ by $ \left( x-9\right) $ $$ \color{blue}{x} \cdot \left( x-9\right) = x^2-9x $$ |
| ② | Multiply $x^2-9x$ by $ \dfrac{x+4}{6} $ to get $ \dfrac{x^3-5x^2-36x}{6} $. Step 1: Write $ x^2-9x $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} x^2-9x \cdot \frac{x+4}{6} & \xlongequal{\text{Step 1}} \frac{x^2-9x}{\color{red}{1}} \cdot \frac{x+4}{6} \xlongequal{\text{Step 2}} \frac{ \left( x^2-9x \right) \cdot \left( x+4 \right) }{ 1 \cdot 6 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^3+4x^2-9x^2-36x }{ 6 } = \frac{x^3-5x^2-36x}{6} \end{aligned} $$ |
| ③ | Multiply $ \dfrac{x^3-5x^2-36x}{6} $ by $ x^3 $ to get $ \dfrac{ x^6-5x^5-36x^4 }{ 6 } $. Step 1: Write $ x^3 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^3-5x^2-36x}{6} \cdot x^3 & \xlongequal{\text{Step 1}} \frac{x^3-5x^2-36x}{6} \cdot \frac{x^3}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^3-5x^2-36x \right) \cdot x^3 }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^6-5x^5-36x^4 }{ 6 } \end{aligned} $$ |
| ④ | Multiply $ \dfrac{x^6-5x^5-36x^4}{6} $ by $ x+4 $ to get $ \dfrac{x^7-x^6-56x^5-144x^4}{6} $. Step 1: Write $ x+4 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^6-5x^5-36x^4}{6} \cdot x+4 & \xlongequal{\text{Step 1}} \frac{x^6-5x^5-36x^4}{6} \cdot \frac{x+4}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ \left( x^6-5x^5-36x^4 \right) \cdot \left( x+4 \right) }{ 6 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x^7+4x^6-5x^6-20x^5-36x^5-144x^4 }{ 6 } = \frac{x^7-x^6-56x^5-144x^4}{6} \end{aligned} $$ |
| ⑤ | Multiply $ \dfrac{x^7-x^6-56x^5-144x^4}{6} $ by $ x-9 $ to get $ \dfrac{x^8-10x^7-47x^6+360x^5+1296x^4}{6} $. Step 1: Write $ x-9 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{x^7-x^6-56x^5-144x^4}{6} \cdot x-9 & \xlongequal{\text{Step 1}} \frac{x^7-x^6-56x^5-144x^4}{6} \cdot \frac{x-9}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( x^7-x^6-56x^5-144x^4 \right) \cdot \left( x-9 \right) }{ 6 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ x^8-9x^7-x^7+9x^6-56x^6+504x^5-144x^5+1296x^4 }{ 6 } = \\[1ex] &= \frac{x^8-10x^7-47x^6+360x^5+1296x^4}{6} \end{aligned} $$ |